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Transcript
Hindawi Publishing Corporation
International Journal of Combinatorics
Volume 2013, Article ID 595210, 34 pages
http://dx.doi.org/10.1155/2013/595210
Review Article
On Bondage Numbers of Graphs: A Survey with
Some Comments
Jun-Ming Xu
School of Mathematical Sciences, University of Science and Technology of China, Wentsun Wu Key Laboratory of CAS, Hefei,
Anhui 230026, China
Correspondence should be addressed to Jun-Ming Xu; [email protected]
Received 15 December 2012; Revised 20 February 2013; Accepted 11 March 2013
Academic Editor: Chฤฑฬnh T. Hoang
Copyright © 2013 Jun-Ming Xu. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The domination number of a graph G is the smallest number of vertices which dominate all remaining vertices by edges of G. The
bondage number of a nonempty graph G is the smallest number of edges whose removal from G results in a graph with domination
number greater than the domination number of G. The concept of the bondage number was formally introduced by Fink et al. in
1990. Since then, this topic has received considerable research attention and made some progress, variations, and generalizations.
This paper gives a survey on the bondage number, including known results, conjectures, problems, and some comments, also
selectively summarizes other types of bondage numbers.
1. Introduction
For terminology and notation on graph theory not given here,
the reader is referred to Xu [1]. Let ๐บ = (๐‘‰, ๐ธ) be a finite,
undirected, and simple graph. We call |๐‘‰| and |๐ธ| the order
and size of ๐บ and denote them by ๐œ = ๐œ(๐บ) and ๐œ€ = ๐œ€(๐บ),
respectively, unless otherwise specified. Through this paper,
the notations ๐‘ƒ๐‘› , ๐ถ๐‘› , and ๐พ๐‘› always denote a path, a cycle,
and a complete graph of order ๐‘›, respectively, the notation
๐พ๐‘›1 ,๐‘›2 ,...,๐‘›๐‘ก denotes a complete ๐‘ก-partite graph with ๐‘›1 โฉฝ ๐‘›2 โฉฝ
โ‹… โ‹… โ‹… โฉฝ ๐‘›๐‘ก and ๐‘›๐‘ก > 1, ๐พ๐‘ก (๐‘š) = ๐พ๐‘›1 ,๐‘›2 ,...,๐‘›๐‘ก with ๐‘›1 = โ‹… โ‹… โ‹… = ๐‘›๐‘ก =
๐‘š, and ๐พ1,๐‘›โˆ’1 is a star. For two vertices ๐‘ฅ and ๐‘ฆ in a connected
graph ๐บ, we use ๐‘‘๐บ(๐‘ฅ, ๐‘ฆ) to denote the distance between ๐‘ฅ
and ๐‘ฆ in ๐บ.
For a vertex ๐‘ฅ in ๐บ, let ๐‘๐บ(๐‘ฅ) be the open set of neighbors
of ๐‘ฅ and ๐‘๐บ[๐‘ฅ] = ๐‘[๐‘ฅ] = ๐‘๐บ(๐‘ฅ) โˆช {๐‘ฅ} the closed
set of neighbors of ๐‘ฅ. For a subset ๐‘‹ โŠ‚ ๐‘‰(๐บ), ๐‘๐บ(๐‘‹) =
(โˆช๐‘ฅโˆˆ๐‘‹ ๐‘๐บ(๐‘ฅ)) โˆฉ ๐‘‹, and ๐‘๐บ[๐‘‹] = ๐‘๐บ(๐‘‹) โˆช ๐‘‹, where ๐‘‹ =
๐‘‰(๐บ) \ ๐‘‹. Let ๐ธ๐‘ฅ be the set of edges incident with ๐‘ฅ in ๐บ;
that is, ๐ธ๐‘ฅ = {๐‘ฅ๐‘ฆ โˆˆ ๐ธ(๐บ) : ๐‘ฆ โˆˆ ๐‘๐บ(๐‘ฅ)}. We denote the
degree of ๐‘ฅ by ๐‘‘๐บ(๐‘ฅ) = |๐ธ๐‘ฅ |. The maximum and the minimum
degrees of ๐บ are denoted by ฮ”(๐บ) and ๐›ฟ(๐บ), respectively. A
vertex of degree zero is called an isolated vertex. An edge
incident with a vertex of degree one is called a pendant
edge.
The bondage number is an important parameter of graphs
which is based upon the well-known domination number.
A subset ๐‘† โŠ† ๐‘‰(๐บ) is called a dominating set of ๐บ if
๐‘๐บ[๐‘†] = ๐‘‰(๐บ); that is, every vertex ๐‘ฅ in ๐‘† has at least one
neighbor in ๐‘†. The domination number of ๐บ, denoted by ๐›พ(๐บ),
is the minimum cardinality among all dominating sets; that
is,
๐›พ (๐บ) = min {|๐‘†| : ๐‘† โŠ† ๐‘‰ (๐บ) , ๐‘๐บ [๐‘†] = ๐‘‰ (๐บ)} .
(1)
A dominating set ๐‘† is called a ๐›พ-set of ๐บ if |๐‘†| = ๐›พ(๐บ).
The domination is such an important and classic conception that it has become one of the most widely studied topics
in graph theory and also is frequently used to study property
of networks. The domination, with many variations and
generalizations, is now well studied in graph and networks
theory. The early vast literature on domination includes the
bibliography compiled by Hedetniemi and Laskar [2] and
a thorough study of domination appears in the books by
Haynes et al. [3, 4]. However, the problem determining the
domination number for general graphs was early proved to
be NP-complete (see GT2 in Appendix in Garey and Johnson
[5], 1979).
Among various problems related with the domination
number, some focus on graph alterations and their effects
2
International Journal of Combinatorics
on the domination number. Here, we are concerned with
a particular graph alternation, the removal of edges from a
graph.
Graphs with domination numbers changed upon the
removal of an edge were first investigated by Walikar and
Acharya [6] in 1979. A graph is called edge-dominationcritical graph if ๐›พ(๐บ โˆ’ ๐‘’) > ๐›พ(๐บ) for every edge ๐‘’ in ๐บ. The
edge-domination-critical graph was characterized by Bauer
et al. [7] in 1983; that is, a graph is edge-domination-critical
if and only if it is the union of stars.
However, for lots of graphs, the domination number is out
of the range of one-edge removal. It is immediate that ๐›พ(๐ป) โฉพ
๐›พ(๐บ) for any spanning subgraph ๐ป of ๐บ. Every graph ๐บ has
a spanning forest ๐‘‡ with ๐›พ(๐บ) = ๐›พ(๐‘‡), and so, in general, a
graph has a nonempty subset ๐น โŠ† ๐ธ(๐บ) for which ๐›พ(๐บ โˆ’ ๐น) =
๐›พ(๐บ).
Then it is natural for the alternation to be generalized to
the removal of several edges, which is just enough to enlarge
the domination number. That is the idea of the bondage
number.
A measure of the efficiency of a domination in graphs was
first given by Bauer et al. [7] in 1983, who called this measure
as domination line-stability, defined as the minimum number
of lines (i.e., edges) which when removed from ๐บ increases ๐›พ.
In 1990, Fink et al. [8] formally introduced the bondage
number as a parameter for measuring the vulnerability of the
interconnection network under link failure. The minimum
dominating set of sites plays an important role in the network
for it dominates the whole network with the minimum cost.
So we must consider whether its function remains good when
the network is attacked. Suppose that someone such as a
saboteur does not know which sites in the network take part
in the dominating role, but does know that the set of these
special sites corresponds to a minimum dominating set in the
related graph. Then how many links does he has to attack so
that the cost cannot remains the same in order to dominate
the whole network? That minimum number of links is just the
bondage number.
The bondage number ๐‘(๐บ) of a nonempty undirected
graph ๐บ is the minimum number of edges whose removal
from ๐บ results in a graph with larger domination number. The
precise definition of the bondage number is as follows:
๐‘ (๐บ) = min {|๐ต| : ๐ต โŠ† ๐ธ (๐บ) , ๐›พ (๐บ โˆ’ ๐ต) > ๐›พ (๐บ)} .
(2)
Since the domination number of every spanning subgraph of
a nonempty graph ๐บ is at least as great as ๐›พ(๐บ), the bondage
number of a nonempty graph is well defined.
We call such an edge-set ๐ต with ๐›พ(๐บ โˆ’ ๐ต) > ๐›พ(๐บ)
the bondage set and the minimum one the minimum
bondage set. In fact, if ๐ต is a minimum bondage set, then
๐›พ(๐บ โˆ’ ๐ต) = ๐›พ(๐บ) + 1 because the removal of one single edge
cannot increase the domination number by more than one.
If ๐‘(๐บ) does not exist, for example, empty graphs, we define
๐‘(๐บ) = โˆž.
It is quite difficult to compute the exact value of the
bondage number for general graphs since it strongly depends
on the domination number of the graphs. Much work focused
on the bounds on the bondage number as well as the restraints
on some particular classes of graphs or networks. The
purpose of this paper is to give a survey of results and research
methods related to these topics for graphs and digraphs.
For some results and research methods, we will make some
comments to develop our further study.
The rest of the paper is organized as follows. Section 2
gives some preliminary results and complexity. Sections
3 and 4 survey some upper bounds and lower bounds,
respectively. The results for some special classes of graphs
and planar graphs are stated in Sections 5 and 6, respectively.
In Section 7, we introduce some results on crossing number
restraints. In Sections 8 and 9, we are concerned about other
and generalized types of bondage numbers, respectively. In
Section 10, we introduce some results for digraphs. In the
last section, we introduce some results for vertex-transitive
graphs by applying efficient dominating sets.
2. Simplicity and Complexity
As we have known from Introduction, the bondage number
is an important parameter for measuring the stability or
the vulnerability of a domination in a graph or a network.
Our aim is to compute the bondage number for any given
graphs or networks. One has determined the exact value of
the bondage number for some graphs with simple structure.
For arbitrarily given graph, however, it has been proved that
determining its bondage number is NP-hard.
2.1. Exact Values for Ordinary Graphs. We begin our investigation of the bondage number by computing its value for
several well-known classes of graphs with simple structure.
In 1990, Fink et al. [8] proposed the concept of the bondage
number and completely determined the exact values of
bondage numbers of some ordinary graphs, such as complete
graphs, paths, cycles, and complete multipartite graphs.
Theorem 1 (Fink et al. [8], 1990). The exact values of bondage
numbers of the following class of graphs are completely determined:
๐‘›
๐‘ (๐พ๐‘› ) = โŒˆ โŒ‰ ;
2
2 ๐‘–๐‘“ ๐‘› โ‰ก 1 (mod 3) ,
๐‘ (๐‘ƒ๐‘› ) = {
1 ๐‘œ๐‘กโ„Ž๐‘’๐‘Ÿ๐‘ค๐‘–๐‘ ๐‘’,
3 ๐‘–๐‘“ ๐‘› โ‰ก 1 (mod 3) ,
๐‘ (๐ถ๐‘› ) = {
2 ๐‘œ๐‘กโ„Ž๐‘’๐‘Ÿ๐‘ค๐‘–๐‘ ๐‘’,
๐‘ (๐พ๐‘›1 ,๐‘›2 ,...,๐‘›๐‘ก )
(3)
๐‘—
{
๐‘–๐‘“ ๐‘›๐‘— = 1 ๐‘Ž๐‘›๐‘‘ ๐‘›๐‘—+1 โฉพ 2, ๐‘“๐‘œ๐‘Ÿ ๐‘ ๐‘œ๐‘š๐‘’ ๐‘—,
โŒˆ โŒ‰
{
{
2
{
{
{
{
1 โฉฝ ๐‘— < ๐‘ก,
{
= {2๐‘ก โˆ’ 1 ๐‘–๐‘“ ๐‘› = ๐‘› = โ‹… โ‹… โ‹… = ๐‘› = 2,
1
2
๐‘ก
{
{
{
๐‘กโˆ’1
{
{
{
{โˆ‘๐‘›๐‘–
๐‘œ๐‘กโ„Ž๐‘’๐‘Ÿ๐‘ค๐‘–๐‘ ๐‘’.
{๐‘–=1
The complete graph ๐พ๐‘› is the unique (๐‘›โˆ’1)-regular graph
of order ๐‘› โฉพ 2, ๐‘(๐พ๐‘› ) = โŒˆ๐‘›/2โŒ‰ by Theorem 1. The ๐‘ก-partite
International Journal of Combinatorics
3
graph ๐พ๐‘ก (2) with ๐‘ก = ๐‘›/2 is an (๐‘› โˆ’ 2)-regular graph ๐บ of
order ๐‘› โฉพ 2, and not unique, ๐‘(๐บ) = ๐‘› โˆ’ 1 by Theorem 1
for an even integer ๐‘› โฉพ 4. For an (๐‘› โˆ’ 3)-regular graph
๐บ of order ๐‘› โฉพ 4, Hu and Xu [9] obtained the following
result.
Theorem 2 (Hu and Xu [9]). ๐‘(๐บ) = ๐‘› โˆ’ 3 for any (๐‘› โˆ’ 3)regular graph ๐บ of order ๐‘› โฉพ 4.
Up to the present, no results have been known for ๐‘˜regular graphs with 3 โฉฝ ๐‘˜ โฉฝ ๐‘› โˆ’ 4. For general graphs, there
are the following two results.
Theorem 3 (Teschner [10], 1997). If ๐บ is a nonempty graph
with a unique minimum dominating set, then ๐‘(๐บ) = 1.
Theorem 4 (Bauer et al. [7], 1983). If any vertex of a graph ๐บ is
adjacent with two or more vertices of degree one, then ๐‘(๐บ) = 1.
Bauer et al. [7] observed that the star is the unique
graph with the property that the bondage number is 1 and
the deletion of any edge results in the domination number
increasing. Motivated by this fact, Hartnell and Rall [11]
proposed the concept of uniformly bonded graphs. A graph is
called to be uniformly bonded if it has bondage number ๐‘ and
the deletion of any ๐‘ edges results in a graph with increased
domination number. Unfortunately, there are a few uniformly
bonded graphs.
Theorem 5 (Hartnell and Rall [11], 1999). The only uniformly
bonded graphs with bondage number 2 are ๐ถ3 and ๐‘ƒ4 . The
unique uniformly bonded graph with bondage number 3 is ๐ถ4 .
There are no such graphs for bondage number greater than 3.
As we mentioned, to compute the exact value of bondage
number for a graph strongly depends upon its domination
number. In this sense, studying the bondage number can
greatly inspire oneโ€™s research concerned with dominations.
However, determining the exact value of domination number
for a given graph is quite difficult. In fact, even if the
exact value of the domination number for some graph is
determined, it is still very difficulty to compute the value
of the bondage number for that graph. For example, for the
hypercube ๐‘„๐‘› , we have ๐›พ(๐‘„๐‘› ) = 2๐‘›โˆ’1 , but we have not yet
determined ๐‘(๐‘„๐‘› ) for any ๐‘› โฉพ 2.
Perhaps Theorems 3 and 4 provide an approach to
compute the exact value of bondage number for some graphs
by establishing some sufficient conditions for ๐‘(๐บ) = ๐‘.
In fact, we will see later that Theorem 3 plays an important
role in determining the exact values of the bondage numbers
for some graphs. Thus, to study the bondage number, it is
important to present various characterizations of graphs with
a unique minimum dominating set.
2.2. Characterizations of Trees. For trees, Bauer et al. [7] in
1983 from the point of view of the domination line-stability,
independently and Fink et al. [8] in 1990 from the point
of view of the domination edge-vulnerability obtained the
following result.
๐‘ค
Figure 1: Tree ๐น๐‘ก .
Theorem 6. For any nontrivial tree ๐‘‡, ๐‘(๐‘‡) โฉฝ 2.
By Theorem 6, it is natural to classify all trees according
to their bondage numbers. Fink et al. [8] proved that a
forbidden subgraph characterization to classify trees with
different bondage numbers is impossible, since they proved
that if ๐น is a forest, then ๐น is an induced subgraph of a tree ๐‘‡
with ๐‘(๐‘‡) = 1 and a tree ๐‘‡๓ธ€  with ๐‘(๐‘‡๓ธ€  ) = 2. However, they
pointed out that the complexity of calculating the bondage
number of a tree of order ๐‘› is at most ๐‘‚(๐‘›2 ) by methodically
removing each pair of edges.
Even so, some characterizations, whether a tree has
bondage number 1 or 2, have been found by several authors;
see, for example, [10, 12, 13].
First we describe the method due to Hartnell and Rall
[12], by which all trees with bondage number 2 can be constructed inductively. An important tree ๐น๐‘ก in the construction
is shown in Figure 1. To characterize this construction, we
need some terminologies:
(1) attach a path ๐‘ƒ๐‘› to a vertex ๐‘ฅ of a tree, which means
to link ๐‘ฅ and one end-vertex of the ๐‘ƒ๐‘› by an edge.
(2) attach ๐น๐‘ก to a vertex ๐‘ฅ, which means to link ๐‘ฅ and a
vertex ๐‘ฆ of ๐น๐‘ก by an edge.
The following are four operations on a tree ๐‘‡:
Type 1: attach a ๐‘ƒ2 to ๐‘ฅ โˆˆ ๐‘‰(๐‘‡), where ๐›พ(๐‘‡ โˆ’ ๐‘ฅ) = ๐›พ(๐‘‡) and ๐‘ฅ
belongs to at least one ๐›พ-set of ๐‘‡ (such a vertex exists,
say, one end-vertex of ๐‘ƒ5 ).
Type 2: attach a ๐‘ƒ3 to ๐‘ฅ โˆˆ ๐‘‰(๐‘‡), where ๐›พ(๐‘‡ โˆ’ ๐‘ฅ) < ๐›พ(๐‘‡).
Type 3: attach ๐น1 to ๐‘ฅ โˆˆ ๐‘‰(๐‘‡), where ๐‘ฅ belongs to at least one
๐›พ-set of ๐‘‡.
Type 4: attach ๐น๐‘ก , ๐‘ก โฉพ 2, to ๐‘ฅ โˆˆ ๐‘‰(๐‘‡), where ๐‘ฅ can be any
vertex of ๐‘‡.
Let C = {๐‘‡ : ๐‘‡ is a tree, and ๐‘‡ = ๐พ1 , ๐‘‡ = ๐‘ƒ4 , and ๐‘‡ = ๐น๐‘ก
for some ๐‘ก โฉพ 2, or ๐‘‡ can be obtained from ๐‘ƒ4 or ๐น๐‘ก (๐‘ก โฉพ 2) by
a finite sequence of operations of Types 1, 2, 3, 4}.
Theorem 7 (Hartnell and Rall [12], 1992). A tree has bondage
number 2 if and only if it belongs to C.
Looking at different minimum dominating sets of a tree,
Teschner [10] presented a totally different characterization
of the set of trees having bondage number 1. They defined
a vertex to be universal if it belongs to each minimum
dominating set, and to be idle if it does not belong to any
minimum dominating set.
4
International Journal of Combinatorics
Theorem 8 (Teschner [10], 1997). A tree ๐‘‡ has bondage
number 1 if and only if ๐‘‡ has a universal vertex or an edge
๐‘ฅ๐‘ฆ satisfying
(1) ๐‘ฅ and ๐‘ฆ are neither universal nor idle; and
(2) all neighbors of ๐‘ฅ and ๐‘ฆ (except for ๐‘ฅ and ๐‘ฆ) are idle.
For a positive integer ๐‘˜, a subset ๐ผ โŠ† ๐‘‰(๐บ) is called a ๐‘˜independent set (also called a ๐‘˜-packing) if ๐‘‘๐บ(๐‘ฅ, ๐‘ฆ) > ๐‘˜ for
any two distinct vertices ๐‘ฅ and ๐‘ฆ in ๐ผ. When ๐‘˜ = 1, 1-set is the
normal independent set. The maximum cardinality among all
๐‘˜-independent sets is called the ๐‘˜-independence number (or
๐‘˜-packing number) of ๐บ, denoted by ๐›ผ๐‘˜ (๐บ). A ๐‘˜-independent
set ๐ผ is called an ๐›ผ๐‘˜ -set if |๐ผ| = ๐›ผ๐‘˜ (๐บ). A graph ๐บ is said to be
๐›ผ๐‘˜ -stable if ๐›ผ๐‘˜ (๐บ) = ๐›ผ๐‘˜ (๐บ โˆ’ ๐‘’) for every edge ๐‘’ of ๐บ. There are
two important results on ๐‘˜-independent sets.
Proposition 9 (Topp and Vestergaard [13], 2000). A tree ๐‘‡ is
๐›ผ๐‘˜ -stable if and only if ๐‘‡ has a unique ๐›ผ๐‘˜ -set.
Proposition 10 (Meir and Moon [14], 1975). ๐›ผ2 (๐บ) โฉฝ ๐›พ(๐บ)
for any connected graph ๐บ with equality for any tree.
Hartnell et al. [15], independently and Topp and Vestergaard [13] also gave a constructive characterization of trees
with bondage number 2 and, applying Proposition 10, presented another characterization of those trees.
Theorem 11 (Hartnell et al. [15], 1998; Top and Vestergaard
[13], 2000). ๐‘(๐‘‡) = 2 for a tree ๐‘‡ if and only if ๐‘‡ has a unique
๐›ผ2 -set.
According to this characterization, Hartnell et al. [15]
presented a linear algorithm for determining the bondage
number of a tree.
In this subsection, we introduce three characterizations
for trees with bondage number 1 or 2. The characterization
in Theorem 7 is constructive, constructing all trees with
bondage number 2, a natural and straightforward method,
by a series of graph-operations. The characterization in
Theorem 8 is a little advisable, by describing the inherent
property of trees with bondage number 1. The characterization in Theorem 11 is wonderful, by using a strong graphtheoretic concept, ๐›ผ๐‘˜ -set. In fact, this characterization is a
byproduct of some results related to ๐›ผ๐‘˜ -sets for trees. It is
that this characterization closed the relation between two
concepts, the bondage number and the ๐‘˜-independent set,
and hence is of research value and important significance.
2.3. Complexity for General Graphs. As mentioned above, the
bondage number of a tree can be determined by a linear time
algorithm. According to this algorithm, we can determine
within polynomial time the domination number of any tree
by removing each edge and verifying whether the domination
number is enlarged according to the known linear time
algorithm for domination numbers of trees.
However, it is impossible to find a polynomial time
algorithm for bondage numbers of general graphs. If such
an algorithm ๐ด exists, then the domination number of any
nonempty undirected graph ๐บ can be determined within
polynomial time by repeatedly using ๐ด. Let ๐บ0 = ๐บ and
๐บ๐‘–+1 = ๐บ๐‘– โˆ’ ๐ต๐‘– , where ๐ต๐‘– is the minimum edge set of ๐บ๐‘– found
by ๐ด such that ๐›พ(๐บ๐‘– โˆ’ ๐ต๐‘– ) = ๐›พ(๐บ๐‘–โˆ’1 ) + 1 for each ๐‘– = 0, 1, . . .;
we can always find the minimum ๐ต๐‘– whose removal from ๐บ๐‘–
enlarges the domination number, until ๐บ๐‘˜ = ๐บ๐‘˜โˆ’1 โˆ’ ๐ต๐‘˜โˆ’1
is empty for some ๐‘˜ โฉพ 1, though ๐ต๐‘˜โˆ’1 is not empty. Then
๐›พ(๐บ) = ๐›พ(๐บ๐‘˜ ) โˆ’ ๐‘˜ = ๐œ(๐บ) โˆ’ ๐‘˜. As known to all, if NP =ฬธ ๐‘ƒ,
the minimum dominating set problem is NP-complete, and
so polynomial time algorithms for the bondage number do
not exist unless NP = ๐‘ƒ.
In fact, Hu and Xu [16] have recently shown that the
problem determining the bondage number of general graphs
is NP-hard.
Problem 1. Consider the decision problem:
Bondage Problem
Instance: a graph ๐บ and a positive integer ๐‘ (โฉฝ ๐œ€(๐บ)).
Question: is ๐‘(๐บ) โฉฝ ๐‘?
Theorem 12 (Hu and Xu [16], 2012). The bondage problem is
NP-hard.
The basic way of the proof is to follow Garey and Johnsonโ€™s
techniques for proving NP-hardness [5] by describing a
polynomial transformation from the known NP-complete
problem: 3-satisfiability problem.
Theorem 12 shows that we are unable to find a polynomial
time algorithm to determine bondage numbers of general
graphs unless NP = ๐‘ƒ. At the same time, this result also
shows that the following study on the bondage number is of
important significance.
(i) Find approximation polynomial algorithms with performance ratio as small as possible.
(ii) Find the lower and upper bounds with difference as
small as possible.
(iii) Determine exact values for some graphs, specially
well-known networks.
Unfortunately, we cannot prove whether or not determining the bondage number is NP-problem since for any subset
๐ต โŠ‚ ๐ธ(๐บ) it is not clear that there is a polynomial algorithm
to verify ๐›พ(๐บ โˆ’ ๐ต) > ๐›พ(๐บ). Since the problem of determining
the domination number is NP-complete, we conjecture that
it is not in NP. This is a worthwhile task to be studied further.
Motivated by the linear time algorithm of Hartnell et al.
to compute the bondage number of a tree, we can make
an attempt to consider whether there is a polynomial time
algorithm to compute the bondage number for some special
classes of graphs such as planar graphs, Cayley graphs, or
graphs with some restrictions of graph-theoretical parameters such as degree, diameter, connectivity, and domination
number.
3. Upper Bounds
By Theorem 12, since we cannot find a polynomial time algorithm for determining the exact values of bondage numbers
International Journal of Combinatorics
5
of general graphs, it is weightily significative to establish
some sharp bounds on the bondage number of a graph. In
this section, we survey several known upper bounds on the
bondage number in terms of some other graph-theoretical
parameters.
๐‘‡2 (๐‘ฅ, ๐‘ฆ)
๐‘‡3 (๐‘ฅ, ๐‘ฆ)
3.1. Most Basic Upper Bounds. We start this subsection with
a simple observation.
๐‘ฆ
Observation 1 (Teschner [10], 1997). Let ๐ป be a spanning
subgraph obtained from a graph ๐บ by removing ๐‘˜ edges. Then
๐‘(๐บ) โฉฝ ๐‘(๐ป) + ๐‘˜.
If we select a spanning subgraph ๐ป such that ๐‘(๐ป) = 1,
then Observation 1 yields some upper bounds on the bondage
number of a graph ๐บ. For example, take ๐ป = ๐บ โˆ’ ๐ธ๐‘ฅ + ๐‘’ and
๐‘˜ = ๐‘‘๐บ(๐‘ฅ) โˆ’ 1, where ๐‘’ โˆˆ ๐ธ๐‘ฅ , the following result can be
obtained.
๐‘‡4 (๐‘ฅ, ๐‘ฆ)
๐‘‡1 (๐‘ฅ, ๐‘ฆ)
๐‘ฅ
Figure 2: Illustration of ๐‘‡๐‘– (๐‘ฅ, ๐‘ฆ) for ๐‘– = 1, 2, 3, 4.
Theorem 13 (Bauer et al. [7], 1983). If there exists at least one
vertex ๐‘ฅ in a graph ๐บ such that ๐›พ(๐บ โˆ’ ๐‘ฅ) โฉพ ๐›พ(๐บ), then ๐‘(๐บ) โฉฝ
๐‘‘๐บ(๐‘ฅ) โฉฝ ฮ”(๐บ).
Theorem 18 (Hartnell and Rall [17], 1994). ๐‘(๐บ) โฉฝ ๐‘‘๐บ(๐‘ฅ) +
๐‘‘๐บ(๐‘ฆ) โˆ’ 1 โˆ’ |๐‘๐บ(๐‘ฅ) โˆฉ ๐‘๐บ(๐‘ฆ)| for any two adjacent vertices ๐‘ฅ
and ๐‘ฆ in a graph ๐บ; that is,
The following early result obtained can be derived from
Observation 1 by taking ๐ป = ๐บ โˆ’ ๐ธ๐‘ฅ๐‘ฆ and ๐‘˜ = ๐‘‘(๐‘ฅ) + ๐‘‘(๐‘ฆ) โˆ’ 2.
๓ต„จ
๓ต„จ
๐‘ (๐บ) โฉฝ min {๐‘‘๐บ (๐‘ฅ) + ๐‘‘๐บ (๐‘ฆ) โˆ’ 1 โˆ’ ๓ต„จ๓ต„จ๓ต„จ๐‘๐บ (๐‘ฅ) โˆฉ ๐‘๐บ (๐‘ฆ)๓ต„จ๓ต„จ๓ต„จ} .
Theorem 14 (Bauer et al. [7], 1983; Fink et al. [8], 1990).
๐‘(๐บ) โฉฝ ๐‘‘(๐‘ฅ) + ๐‘‘(๐‘ฆ) โˆ’ 1 for any two adjacent vertices ๐‘ฅ and
๐‘ฆ in a graph ๐บ; that is,
๐‘ (๐บ) โฉฝ min {๐‘‘๐บ (๐‘ฅ) + ๐‘‘๐บ (๐‘ฆ) โˆ’ 1} .
๐‘ฅ๐‘ฆโˆˆ๐ธ(๐บ)
(4)
This theorem gives a natural corollary obtained by several
authors.
Corollary 15 (Bauer et al. [7], 1983; Fink et al. [8], 1990). If ๐บ
is a graph without isolated vertices, then ๐‘(๐บ) โฉฝ ฮ”(๐บ)+๐›ฟ(๐บ)โˆ’1.
In 1999, Hartnell and Rall [11] extended Theorem 14 to
the following more general case, which can be also derived
from Observation 1 by taking ๐ป = ๐บ โˆ’ ๐ธ๐‘ฅ โˆ’ ๐ธ๐‘ฆ + (๐‘ฅ, ๐‘ง, ๐‘ฆ) if
๐‘‘๐บ(๐‘ฅ, ๐‘ฆ) = 2, where (๐‘ฅ, ๐‘ง, ๐‘ฆ) is a path of length 2 in ๐บ.
Theorem 16 (Hartnell and Rall [11], 1999). ๐‘(๐บ) โฉฝ ๐‘‘(๐‘ฅ) +
๐‘‘(๐‘ฆ) โˆ’ 1 for any distinct two vertices ๐‘ฅ and ๐‘ฆ in a graph ๐บ with
๐‘‘๐บ(๐‘ฅ, ๐‘ฆ) โฉฝ 2; that is,
๐‘ (๐บ) โฉฝ min {๐‘‘๐บ (๐‘ฅ) + ๐‘‘๐บ (๐‘ฆ) โˆ’ 1} .
๐‘‘๐บ (๐‘ฅ,๐‘ฆ)โฉฝ2
(5)
๐‘ฅ๐‘ฆโˆˆ๐ธ(๐บ)
(6)
These results give simple but important upper bounds on
the bondage number of a graph, and is also the foundation
of almost all results on bondage numbers upper bounds
obtained till now.
By careful consideration of the nature of the edges from
the neighbors of ๐‘ฅ and ๐‘ฆ, Wang [18] further refined the bound
in Theorem 18. For any edge ๐‘ฅ๐‘ฆ โˆˆ ๐ธ(๐บ), ๐‘๐บ(๐‘ฆ) contains the
following four subsets:
(1) ๐‘‡1 (๐‘ฅ, ๐‘ฆ) = ๐‘๐บ[๐‘ฅ] โˆฉ ๐‘๐บ(๐‘ฆ);
(2) ๐‘‡2 (๐‘ฅ, ๐‘ฆ) = {๐‘ค โˆˆ ๐‘๐บ(๐‘ฆ) : ๐‘๐บ(๐‘ค) โŠ† ๐‘๐บ(๐‘ฆ) โˆ’ ๐‘ฅ};
(3) ๐‘‡3 (๐‘ฅ, ๐‘ฆ) = {๐‘ค โˆˆ ๐‘๐บ(๐‘ฆ) : ๐‘๐บ(๐‘ค) โŠ† ๐‘๐บ(๐‘ง)โˆ’๐‘ฅ for some
๐‘ง โˆˆ ๐‘๐บ(๐‘ฅ) โˆฉ ๐‘๐บ(๐‘ฆ)};
(4) ๐‘‡4 (๐‘ฅ, ๐‘ฆ) = ๐‘๐บ(๐‘ฆ) \ (๐‘‡1 (๐‘ฅ, ๐‘ฆ) โˆช ๐‘‡2 (๐‘ฅ, ๐‘ฆ) โˆช ๐‘‡3 (๐‘ฅ, ๐‘ฆ)).
The illustrations of ๐‘‡1 (๐‘ฅ, ๐‘ฆ), ๐‘‡2 (๐‘ฅ, ๐‘ฆ), ๐‘‡3 (๐‘ฅ, ๐‘ฆ), and
๐‘‡4 (๐‘ฅ, ๐‘ฆ) are shown in Figure 2 (corresponding vertices
pointed by dashed arrows).
Theorem 19 (Wang [18], 1996). For any nonempty graph ๐บ,
Corollary 17 (Fink et al. [8], 1990). If a vertex of a graph ๐บ is
adjacent with two or more vertices of degree one, then ๐‘(๐บ) = 1.
๓ต„จ
๓ต„จ
๐‘ (๐บ) โฉฝ min {๐‘‘๐บ (๐‘ฅ) + ๓ต„จ๓ต„จ๓ต„จ๐‘‡4 (๐‘ฅ, ๐‘ฆ)๓ต„จ๓ต„จ๓ต„จ} .
We remark that the bounds stated in Corollary 15 and
Theorem 16 are sharp. As indicated by Theorem 1, one class of
graphs in which the bondage number achieves these bounds
is the class of cycles whose orders are congruent to 1 modulo
3.
On the other hand, Hartnell and Rall [17] sharpened the
upper bound in Theorem 14 as follows, which can be also
derived from Observation 1.
The graph in Figure 2 shows that the upper bound given
in Theorem 19 is better than that in Theorems 16 and 18,
for the upper bounds obtained from these two theorems are
๐‘‘๐บ(๐‘ฅ)+๐‘‘๐บ(๐‘ฆ)โˆ’1 = 11 and ๐‘‘๐บ(๐‘ฅ)+๐‘‘๐บ(๐‘ฆ)โˆ’|๐‘๐บ(๐‘ฅ)โˆฉ๐‘๐บ(๐‘ฆ)| =
9, respectively, while the upper bound given by Theorem 19 is
๐‘‘๐บ(๐‘ฅ) + |๐‘‡4 (๐‘ฅ, ๐‘ฆ)| = 6.
The following result is also an improvement of
Theorem 14, in which ๐‘ก = 2.
๐‘ฅ๐‘ฆโˆˆ๐ธ(๐บ)
(7)
6
International Journal of Combinatorics
Theorem 20 (Teschner [10], 1997). If ๐บ contains a complete
subgraph ๐พ๐‘ก with ๐‘ก โฉพ 2, then ๐‘(๐บ) โฉฝ min๐‘ฅ๐‘ฆโˆˆ๐ธ(๐พ๐‘ก ) {๐‘‘๐บ(๐‘ฅ) +
๐‘‘๐บ(๐‘ฆ) โˆ’ ๐‘ก + 1}.
Following Fricke et al. [19], a vertex ๐‘ฅ of a graph ๐บ is ๐›พgood if ๐‘ฅ belongs to some ๐›พ-set of ๐บ and ๐›พ-bad if ๐‘ฅ belongs to
no ๐›พ-set of ๐บ. Let ๐ด(๐บ) be the set of ๐›พ-good vertices, and let
๐ต(๐บ) be the set of ๐›พ-bad vertices in ๐บ. Clearly, {๐ด(๐บ), ๐ต(๐บ)}
is a partition of ๐‘‰(๐บ). Note there exists some ๐‘ฅ โˆˆ ๐ด such that
๐›พ(๐บ โˆ’ ๐‘ฅ) = ๐›พ(๐บ), say, one end-vertex of ๐‘ƒ5 . Samodivkin [20]
presented some sharp upper bounds for ๐‘(๐บ) in terms of ๐›พgood and ๐›พ-bad vertices of ๐บ.
๐ป2
๐‘ฅ22
๐‘ฅ21
๐‘ฅ12
๐ป1
๐ป3
๐‘ฅ3
๐‘ฅ
Theorem 21 (Samodivkin [20], 2008). Let ๐บ be a graph.
๐‘ฅ11
(a) Let ๐ถ(๐บ) = {๐‘ฅ โˆˆ ๐‘‰(๐บ) : ๐›พ(๐บ โˆ’ ๐‘ฅ) โฉพ ๐›พ(๐บ)}. If ๐ถ(๐บ) =ฬธ 0,
then
๐‘ (๐บ) โฉฝ min {๐‘‘๐บ (๐‘ฅ) + ๐›พ (๐บ) โˆ’ ๐›พ (๐บ โˆ’ ๐‘ฅ) : ๐‘ฅ โˆˆ ๐ถ (๐บ)} .
(8)
Figure 3: The graph ๐บ2 .
๐‘ฅ๐‘–1 , ๐‘ฅ๐‘–2 โˆˆ ๐‘‰(๐ป๐‘– ), ๐‘ฅ๐‘–1 ๐‘ฅ๐‘–2 โˆ‰ ๐ธ(๐ป๐‘– ) for each ๐‘– = 1, 2, . . . , ๐‘ก. The
graph ๐บ๐‘ก is defined as follows:
๐‘ก+1
(b) If ๐ต =ฬธ 0, then
๓ต„จ
๓ต„จ
๐‘ (๐บ) โฉฝ min {๓ต„จ๓ต„จ๓ต„จ๐‘๐บ (๐‘ฅ) โˆฉ ๐ด๓ต„จ๓ต„จ๓ต„จ : ๐‘ฅ โˆˆ ๐ต (๐บ)} .
๐‘‰ (๐บ๐‘ก ) = โ‹ƒ ๐‘‰ (๐ป๐‘˜ ) ,
(9)
Theorem 22 (Samodivkin [20], 2008). Let ๐บ be a graph. If
๐ด+ (๐บ) = {๐‘ฅ โˆˆ ๐ด(๐บ) : ๐‘‘๐บ(๐‘ฅ) โฉพ 1 and ๐›พ(๐บ โˆ’ ๐‘ฅ) < ๐›พ(๐บ)} =ฬธ 0,
then
๐‘ (๐บ) โฉฝ
min
๐‘ฅโˆˆ๐ด+ (๐บ),๐‘ฆโˆˆ๐ต(๐บโˆ’๐‘ฅ)
๐‘ฆ
๓ต„จ
๓ต„จ
{๐‘‘๐บ (๐‘ฅ) + ๓ต„จ๓ต„จ๓ต„จ๐‘๐บ (๐‘ฆ) โˆฉ ๐ด (๐บ โˆ’ ๐‘ฅ)๓ต„จ๓ต„จ๓ต„จ} .
(10)
Proposition 23 (Samodivkin [20], 2008). Under the notation
of Theorem 19, if ๐‘ฅ โˆˆ ๐ด+ (๐บ), then (๐‘‡1 (๐‘ฅ, ๐‘ฆ) โˆ’ {๐‘ฅ}) โˆช ๐‘‡2 (๐‘ฅ, ๐‘ฆ) โˆช
๐‘‡3 (๐‘ฅ, ๐‘ฆ) โŠ† ๐‘๐บ(๐‘ฆ) \ ๐ต(๐บ โˆ’ ๐‘ฅ).
By Proposition 23, if ๐‘ฅ โˆˆ ๐ด+ (๐บ), then
๐‘ก+1
๐‘ก
๐‘˜=0
๐‘–=1
(12)
๐ธ (๐บ๐‘ก ) = ( โ‹ƒ ๐ธ (๐ป๐‘˜ ))โˆช(โ‹ƒ {๐‘ฅ๐‘ฅ๐‘–1 , ๐‘ฅ๐‘ฅ๐‘–2 })โˆช{๐‘ฅ๐‘ฅ๐‘ก+1 } .
Such a constructed graph ๐บ๐‘ก is shown in Figure 3 when ๐‘ก = 2.
Observe that ๐›พ(๐บ๐‘ก ) = ๐‘ก + 2, ๐ด(๐บ๐‘ก ) = ๐‘‰(๐บ๐‘ก ), ๐‘‘๐บ๐‘ก (๐‘ฅ) =
2๐‘ก + 2, ๐‘‘๐บ๐‘ก (๐‘ฅ๐‘ก+1 ) = ๐‘ก + 3, ๐‘‘๐บ๐‘ก (๐‘ฆ) = 1, and ๐‘‘๐บ๐‘ก (๐‘ง) = ๐‘ก + 2
for each ๐‘ง โˆˆ ๐‘‰(๐บ๐‘ก โˆ’ {๐‘ฅ, ๐‘ฆ, ๐‘ฅ๐‘ก+1 }). Moreover, ๐›พ(๐บ โˆ’ ๐‘ฆ) < ๐›พ(๐บ)
and ๐›พ(๐บ๐‘ก โˆ’ ๐‘ง) = ๐›พ(๐บ๐‘ก ) for any ๐‘ง โˆˆ ๐‘‰(๐บ๐‘ก ) โˆ’ {๐‘ฆ}. Hence, each of
the bounds stated in Theorems 13โ€“20 is greater than or equals
๐‘ก + 2.
Consider the graph ๐บ๐‘ก โˆ’ ๐‘ฅ๐‘ฆ. Clearly, ๐›พ(๐บ๐‘ก โˆ’ ๐‘ฅ๐‘ฆ) = ๐›พ(๐บ๐‘ก )
and
๐ต (๐บ๐‘ก โˆ’ ๐‘ฅ๐‘ฆ) = ๐ต (๐บ๐‘ก โˆ’ ๐‘ฆ)
๐‘ก
= {๐‘ฅ} โˆช ๐‘‰ (๐ป๐‘ก+1 โˆ’ ๐‘ฅ๐‘ก+1 ) โˆช ( โ‹ƒ {๐‘ฅ๐‘˜1 , ๐‘ฅ๐‘˜2 }) .
๓ต„จ
๓ต„จ
๐‘‘๐บ (๐‘ฅ) + min {๓ต„จ๓ต„จ๓ต„จ๐‘‡4 (๐‘ฅ, ๐‘ฆ)๓ต„จ๓ต„จ๓ต„จ}
๐‘ฆโˆˆ๐‘ (๐‘ฅ)
๐‘˜=1
๐บ
๓ต„จ
๓ต„จ
โฉพ ๐‘‘๐บ (๐‘ฅ) + min {๓ต„จ๓ต„จ๓ต„จ๐‘๐บ (๐‘ฆ) โˆฉ ๐ด (๐บ โˆ’ ๐‘ฅ)๓ต„จ๓ต„จ๓ต„จ}
๐‘ฆโˆˆ๐‘๐บ (๐‘ฅ)
๐‘˜=0
(11)
๓ต„จ
๓ต„จ
โฉพ ๐‘‘๐บ (๐‘ฅ) + min {๓ต„จ๓ต„จ๓ต„จ๐‘๐บ (๐‘ฆ) โˆฉ ๐ด (๐บ โˆ’ ๐‘ฅ)๓ต„จ๓ต„จ๓ต„จ} .
๐‘ฆโˆˆ๐ต(๐บโˆ’๐‘ฅ)
Hence, Theorem 19 can be seen to follow from Theorem 22.
Any graph ๐บ with ๐‘(๐บ) achieving the upper bound in
Theorem 19 can be used to show that the bound in
Theorem 22 is sharp.
Let ๐‘ก โฉพ 2 be an integer. Samodivkin [20] constructed a very interesting graph ๐บ๐‘ก to show that the upper
bound in Theorem 22 is better than the known bounds. Let
๐ป0 , ๐ป1 , ๐ป2 , . . . , ๐ป๐‘ก+1 be mutually vertex-disjoint graphs such
that ๐ป0 โ‰… ๐พ2 , ๐ป๐‘ก+1 โ‰… ๐พ๐‘ก+3 and ๐ป๐‘– โ‰… ๐พ๐‘ก+3 โˆ’ ๐‘’ for each
๐‘– = 1, 2, . . . , ๐‘ก. Let ๐‘‰(๐ป0 ) = {๐‘ฅ, ๐‘ฆ}, ๐‘ฅ๐‘ก+1 โˆˆ ๐‘‰(๐ป๐‘ก+1 ) and
(13)
Therefore, ๐‘๐บ๐‘ก (๐‘ฅ) โˆฉ ๐บ(๐บ๐‘ก โˆ’ ๐‘ฆ) = {๐‘ฅ๐‘ก+1 } which implies that
the upper bound stated in Theorem 22 is equal to ๐‘‘๐บ๐‘ก (๐‘ฆ) +
|{๐‘ฅ๐‘ก+1 }| = 2. Clearly, ๐‘(๐บ๐‘ก ) = 2, and hence this bound is sharp
for ๐บ๐‘ก .
From the graph ๐บ๐‘ก , we obtain the following statement
immediately.
Proposition 24. For every integer ๐‘ก โฉพ 2, there is a graph ๐บ
such that the difference between any upper bound stated in
Theorems 13โ€“20 and the upper bound in Theorem 22 is equal
to ๐‘ก.
Although Theorem 22 supplies us with the upper bound
that is closer to ๐‘(๐บ) for some graph ๐บ than what any one of
International Journal of Combinatorics
Theorems 13โ€“20 provides, it is not easy to determine the sets
๐ด+ (๐บ) and ๐ต(๐บ) mentioned in Theorem 22 for an arbitrary
graph ๐บ. Thus, the upper bound given in Theorem 22 is of
theoretical importance, but not applied since, until now, we
have not found a new class of graphs whose bondage numbers
are determined by Theorem 22.
The above-mentioned upper bounds on the bondage
number are involved in only degrees of two vertices. Hartnell
and Rall [11] established an upper bound on ๐‘(๐บ) in terms of
the numbers of vertices and edges of ๐บ with order ๐‘›. For any
connected graph ๐บ, let ๐›ฟ(๐บ) represent the average degree of
vertices in ๐บ; that is, ๐›ฟ(๐บ) = (1/๐‘›) โˆ‘๐‘ฅโˆˆ๐‘‰ ๐‘‘๐บ(๐‘ฅ). Hartnell and
Rall first discovered the following proposition.
Proposition 25. For any connected graph ๐บ, there exist two
vertices ๐‘ฅ and ๐‘ฆ with distance at most two and with the
property that ๐‘‘๐บ(๐‘ฅ) + ๐‘‘๐บ(๐‘ฆ) โฉฝ 2๐›ฟ(๐บ).
Using Proposition 25 and Theorem 16, Hartnell and Rall
gave the following bound.
Theorem 26 (Hartnell and Rall [11], 1999). ๐‘(๐บ) โฉฝ 2๐›ฟ(๐บ) โˆ’ 1
for any connected graph ๐บ.
Note that 2๐œ€ = ๐‘›๐›ฟ(๐บ) for any graph ๐บ with order ๐‘› and
size ๐œ€. Theorem 26 implies the following bound in terms of
order ๐‘› and size ๐œ€.
Corollary 27. ๐‘(๐บ) โฉฝ (4๐œ€/๐‘›) โˆ’ 1 for any connected graph ๐บ
with order ๐‘› and size ๐œ€.
A lower bound on ๐œ€ in terms of order ๐‘› and the bondage
number is obtained from Corollary 28 immediately.
Corollary 28. ๐œ€ โฉพ (๐‘›/4)(๐‘(๐บ) + 1) for any connected graph ๐บ
with order ๐‘› and size ๐œ€.
Hartnell and Rall [11] gave some graphs ๐บ with order ๐‘›
to show that for each value of ๐‘(๐บ); the lower bound on ๐œ€(๐บ)
given in the Corollary 28 is sharp for some values of ๐‘›.
3.2. Bounds Implied by Connectivity. Use ๐œ…(๐บ) and ๐œ†(๐บ) to
denote the vertex-connectivity and the edge-connectivity of
a connected graph ๐บ, respectively, which are the minimum
numbers of vertices and edges whose removal result in ๐บ
disconnected. The famous Whitney inequality is stated as
๐œ…(๐บ) โฉฝ ๐œ†(๐บ) โฉฝ ๐›ฟ(๐บ) for any graph or digraph ๐บ. Corollary 15
is improved by several authors as follows.
Theorem 29 (Hartnell and Rall [17], 1994; Teschner [10],
1997). If ๐บ is a connected graph, then ๐‘(๐บ) โฉฝ ๐œ†(๐บ) + ฮ”(๐บ) โˆ’ 1.
The upper bound given in Theorem 29 can be attained.
For example, a cycle ๐ถ3๐‘˜+1 with ๐‘˜ โฉพ 1, ๐‘(๐ถ3๐‘˜+1 ) = 3 by
Theorem 1. Since ๐œ…(๐ถ3๐‘˜+1 ) = ๐œ†(๐ถ3๐‘˜+1 ) = 2, we have ๐œ…(๐ถ3๐‘˜+1 )+
๐œ†(๐ถ3๐‘˜+1 ) โˆ’ 1 = 2 + 2 โˆ’ 1 = 3 = ๐‘(๐ถ3๐‘˜+1 ).
Motivated by Corollary 15, Theorems 29, and the Whitney
inequality, Dunbar et al. [21] naturally proposed the following
conjecture.
7
Figure 4: A graph ๐ป with ๐›พ(๐ป) = 3 and ๐‘(๐ป) = 5.
Conjecture 30. If ๐บ is a connected graph, then ๐‘(๐บ) โฉฝ ฮ”(๐บ) +
๐œ…(๐บ) โˆ’ 1.
However, Liu and Sun [22] presented a counterexample
to this conjecture. They first constructed a graph ๐ป showed
in Figure 4 with ๐›พ(๐ป) = 3 and ๐‘(๐ป) = 5. Then, let ๐บ be the
disjoint union of two copies of ๐ป by identifying two vertices
of degree two. They proved that ๐‘(๐บ) โฉพ 5. Clearly, ๐บ is a 4regular graph with ๐œ…(๐บ) = 1 and ๐œ†(๐บ) = 2, and so ๐‘(๐บ) โฉฝ 5
by Theorem 29. Thus, ๐‘(๐บ) = 5 > 4 = ฮ”(๐บ) + ๐œ…(๐บ) โˆ’ 1.
With a suspicion of the relationship between the bondage
number and the vertex-connectivity of a graph, the following
conjecture is proposed.
Conjecture 31 (Liu and Sun [22], 2003). For any positive
integer ๐‘Ÿ, there exists a connected graph ๐บ such that ๐‘(๐บ) โฉพ
ฮ”(๐บ) + ๐œ…(๐บ) + ๐‘Ÿ.
To the knowledge of the author, until now no results have
been known about this conjecture.
We conclude this subsection with the following remarks.
From Theorem 29, if Conjecture 31 holds for some connected
graph ๐บ, then ๐œ†(๐บ) > ๐œ…(๐บ)+๐‘Ÿ, which implies that ๐บ is of large
edge-connectivity and small vertex-connectivity. Use ๐œ‰(๐บ) to
denote the minimum edge-degree of ๐บ; that is,
๐œ‰ (๐บ) = min {๐‘‘๐บ (๐‘ฅ) + ๐‘‘๐บ (๐‘ฆ) โˆ’ 2} .
๐‘ฅ๐‘ฆโˆˆ๐ธ(๐บ)
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Theorem 14 implies the following result.
Proposition 32. ๐‘(๐บ) โฉฝ ๐œ‰(๐บ) + 1 for any graph ๐บ.
Use ๐œ†๓ธ€  (๐บ) to denote the restricted edge-connectivity of
a connected graph ๐บ, which is the minimum number of
edges whose removal result in ๐บ disconnected and no isolated
vertices.
Proposition 33 (Esfahanian and Hakimi [23], 1988). If ๐บ is
neither ๐พ1,๐‘› nor ๐พ3 , then ๐œ†๓ธ€  (๐บ) โฉฝ ๐œ‰(๐บ).
Combining Propositions 32 and 33, we propose a conjecture.
Conjecture 34. If a connected ๐บ is neither ๐พ1,๐‘› nor ๐พ3 , then
๐‘(๐บ) โฉฝ ๐›ฟ(๐บ) + ๐œ†๓ธ€  (๐บ) โˆ’ 1.
8
A cycle ๐ถ3๐‘˜+1 also satisfies ๐‘(๐ถ3๐‘˜+1 ) = 3 = ๐›ฟ(๐ถ3๐‘˜+1 ) +
๐œ†๓ธ€  (๐ถ3๐‘˜+1 ) โˆ’ 1 since ๐œ†๓ธ€  (๐ถ3๐‘˜+1 ) = ๐›ฟ(๐ถ3๐‘˜+1 ) = 2 for any integer
๐‘˜ โฉพ 1.
For the graph ๐ป shown in Figure 4, ๐œ†๓ธ€  (๐ป) = 4 and ๐›ฟ(๐ป) =
2, and so ๐‘(๐ป) = 5 = ๐›ฟ(๐ป) + ๐œ†๓ธ€  (๐ป) โˆ’ 1.
For the 4-regular graph ๐บ constructed by Liu and Sun [22]
obtained from the disjoint union of two copies of the graph
๐ป showed in Figure 4 by identifying two vertices of degree
two, we have ๐‘(๐บ) โฉพ 5. Clearly, ๐œ†๓ธ€  (๐บ) = 2. Thus, ๐‘(๐บ) = 5 =
๐›ฟ(๐บ) + ๐œ†๓ธ€  (๐บ) โˆ’ 1.
For the 4-regular graph ๐บ๐‘ก constructed by Samodivkin
[20], see Figure 3 for ๐‘ก = 2, ๐‘(๐บ๐‘ก ) = 2. Clearly, ๐›ฟ(๐บ๐‘ก ) = 1
and ๐œ†๓ธ€  (๐บ) = 2. Thus, ๐‘(๐บ) = 2 = ๐›ฟ(๐บ) + ๐œ†๓ธ€  (๐บ) โˆ’ 1.
These examples show that if Conjecture 34 is true then the
given upper bound is tight.
3.3. Bounds Implied by Degree Sequence. Now let us return to
Theorem 16, from which Teschner [10] obtained some other
bounds in terms of the degree sequences. The degree sequence
๐œ‹(๐บ) of a graph ๐บ with vertex-set ๐‘‰ = {๐‘ฅ1 , ๐‘ฅ2 , . . . , ๐‘ฅ๐‘› } is the
sequence ๐œ‹ = (๐‘‘1 , ๐‘‘2 , . . . , ๐‘‘๐‘› ) with ๐‘‘1 โฉฝ ๐‘‘2 โฉฝ โ‹… โ‹… โ‹… โฉฝ ๐‘‘๐‘› , where
๐‘‘๐‘– = ๐‘‘๐บ(๐‘ฅ๐‘– ) for each ๐‘– = 1, 2, . . . , ๐‘›. The following result is
essentially a corollary of Theorem 16.
Theorem 35 (Teschner [10], 1997). Let ๐บ be a nonempty graph
with degree sequence ๐œ‹(๐บ). If ๐›ผ2 (๐บ) = ๐‘ก, then ๐‘(๐บ) โฉฝ ๐‘‘๐‘ก +๐‘‘๐‘ก+1 โˆ’
1.
Combining Theorem 35 with Proposition 10 (i.e., ๐›ผ2 (๐บ) โฉฝ
๐›พ(๐บ)), we have the following corollary.
Corollary 36 (Teschner [10], 1997). Let ๐บ be a nonempty
graph with the degree sequence ๐œ‹(๐บ). If ๐›พ(๐บ) = ๐›พ, then ๐‘(๐บ) โฉฝ
๐‘‘๐›พ + ๐‘‘๐›พ+1 โˆ’ 1.
In [10], Teschner showed that these two bounds are sharp
for arbitrarily many graphs. Let ๐ป = ๐ถ3๐‘˜+1 + {๐‘ฅ1 ๐‘ฅ4 , ๐‘ฅ1 ๐‘ฅ3๐‘˜โˆ’1 },
where ๐ถ3๐‘˜+1 is a cycle (๐‘ฅ1 , ๐‘ฅ2 , . . . , ๐‘ฅ3๐‘˜+1 , ๐‘ฅ1 ) for any integer
๐‘˜ โฉพ 2. Then ๐›พ(๐ป) = ๐‘˜ + 1 and so ๐‘(๐ป) โฉฝ 2 + 2 โˆ’ 1 = 3
by Corollary 36. Since ๐ถ3๐‘˜+1 is a spanning subgraph of ๐ป and
๐›พ(๐บ) = ๐›พ(๐ป), Observation 1 yields that ๐‘(๐ป) โฉพ ๐‘(๐ถ3๐‘˜+1 ) = 3,
and so ๐‘(๐ป) = 3.
Although various upper bounds have been established
as the above, we find that the appearance of these bounds
is essentially based upon the local structures of a graph,
precisely speaking, the structures of the neighborhoods of
two vertices within distance 2. Even if these bounds can be
achieved by some special graphs, it is more often not the case.
The reason lies essentially in the definition of the bondage
number, which is the minimum value among all bondage sets,
an integral property of a graph. While it easy to find upper
bounds just by choosing some bondage set, the gap between
the exact value of the bondage number and such a bound
obtained only from local structures of a graph is often large.
For example, a star ๐พ1,ฮ” , however large ฮ” is, ๐‘(๐พ1,ฮ” ) = 1.
Therefore, one has been longing for better bounds upon some
integral parameters. However, as what we will see below, it is
difficult to establish such upper bounds.
International Journal of Combinatorics
3.4. Bounds in ๐›พ-Critical Graphs. A graph ๐บ is called a vertex
domination-critical graph (vc-graph or ๐›พ-critical for short) if
๐›พ(๐บ โˆ’ ๐‘ฅ) < ๐›พ(๐บ) for any ๐‘ฅ โˆˆ ๐‘‰(๐บ), proposed by Brigham
et al. [24] in 1988. Several families of graphs are known to be
๐›พ-critical. From definition, it is clear that if ๐บ is a ๐›พ-critical
graph, then ๐›พ(๐บ) โฉพ 2. The class of ๐›พ-critical graphs with ๐›พ = 2
is characterized as follows.
Proposition 37 (Brigham et al. [24], 1988). A graph ๐บ with
๐›พ(๐บ) = 2 is a ๐›พ-critical graph if and only if ๐บ is a complete
graph ๐พ2๐‘ก (๐‘ก โฉพ 2) with a perfect matching removed.
A more interesting family is composed of the ๐‘›-critical
graphs ๐บ๐‘š,๐‘› defined for ๐‘š, ๐‘› โฉพ 2 by the circulant undirected
graph ๐บ(๐‘, ±๐‘†), where ๐‘ = (๐‘š + 1)(๐‘› โˆ’ 1) + 1 and
๐‘† = {1, 2, . . . , โŒŠ๐‘š/2โŒ‹}, in which ๐‘‰(๐บ(๐‘, ±๐‘†)) = ๐‘ and
๐ธ(๐บ(๐‘, ±๐‘†)) = {๐‘–๐‘— : |๐‘— โˆ’ ๐‘–| โ‰ก ๐‘  (mod ๐‘), ๐‘  โˆˆ ๐‘†}.
The reason why ๐›พ-critical graphs are of special interest in
this context is easy to see that they play an important role in
study of the bondage number. For instance, it immediately
follows from Theorem 13 that if ๐‘(๐บ) > ฮ”(๐บ) then ๐บ is a ๐›พcritical graph. The ๐›พ-critical graphs are defined exactly in this
way. In order to find graphs ๐บ with a high bondage number
(i.e., higher than ฮ”(๐บ)) and beyond its general upper bounds
for the bondage number, we, therefore, have to look at ๐›พcritical graphs.
The bondage numbers of some ๐›พ-critical graphs have
been examined by several authors; see, for example, [10,
20, 25, 26]. From Theorem 13, we know that the bondage
number of a graph ๐บ is bounded from above by ฮ”(๐บ)
if ๐บ is not a ๐›พ-critical graph. For ๐›พ-critical graphs, it is
more difficult to find an upper bound. We will see that the
bondage numbers of ๐›พ-critical graphs in general are not
even bounded from above by ฮ” + ๐‘ for any fixed natural
number ๐‘.
In this subsection, we introduce some upper bounds for
the bondage number of a ๐›พ-critical graph. By Proposition 37,
we easily obtain the following result.
Theorem 38. If ๐บ is a ๐›พ-critical graph with ๐›พ(๐บ) = 2, then
๐‘(๐บ) โฉฝ ฮ” + 1.
In Section 4, by Theorem 59, we will see that the equality
in Theorem 38 holds; that is, ๐‘(๐บ) = ฮ” + 1 if ๐บ is a ๐›พ-critical
graph with ๐›พ(๐บ) = 2.
Theorem 39 (Teschner [10], 1997). Let ๐บ be a ๐›พ-critical graph
with degree sequence ๐œ‹(๐บ). Then ๐‘(๐บ) โฉฝ max{ฮ”(๐บ) + 1, ๐‘‘1 +
๐‘‘2 + โ‹… โ‹… โ‹… + ๐‘‘๐›พโˆ’1 }, where ๐›พ = ๐›พ(๐บ).
As we mentioned above, if ๐บ is a ๐›พ-critical graph with
๐›พ(๐บ) = 2, then ๐‘(๐บ) = ฮ” + 1, which shows that the
bound given in Theorem 39 can be attained for ๐›พ = 2.
However, we have not known whether this bound is tight
for general ๐›พ โฉพ 3. Theorem 39 gives the following corollary
immediately.
Corollary 40 (Teschner [10], 1997). Let ๐บ be a ๐›พ-critical
graph with degree sequence ๐œ‹(๐บ). If ๐›พ(๐บ) = 3, then ๐‘(๐บ) โฉฝ
max{ฮ”(๐บ) + 1, ๐›ฟ(๐บ) + ๐‘‘2 }.
International Journal of Combinatorics
From Theorem 38 and Corollary 40, we have ๐‘(๐บ) โฉฝ
2ฮ”(๐บ) if ๐บ is a ๐›พ-critical graph with ๐›พ(๐บ) โฉฝ 3. The following
result shows that this bound is not tight.
Theorem 41 (Teschner [26], 1995). Let ๐บ be a ๐›พ-critical graph
graph with ๐›พ(๐บ) โฉฝ 3. Then ๐‘(๐บ) โฉฝ (3/2)ฮ”(๐บ).
Until now, we have not known whether the bound given
in Theorem 41 is tight or not. We state two conjectures on
๐›พ-critical graphs proposed by Samodivkin [20]. The first of
them is motivated by Theorems 22 and 19.
Conjecture 42 (Samodivkin [20], 2008). For every connected
nontrivial ๐›พ-critical graph ๐บ,
min
๐‘ฅโˆˆ๐ด+ (๐บ),๐‘ฆโˆˆ๐ต(๐บโˆ’๐‘ฅ)
3
๓ต„จ
๓ต„จ
{๐‘‘๐บ (๐‘ฅ) + ๓ต„จ๓ต„จ๓ต„จ๐‘๐บ (๐‘ฆ) โˆฉ ๐ด (๐บ โˆ’ ๐‘ฅ)๓ต„จ๓ต„จ๓ต„จ} โฉฝ ฮ” (๐บ) .
2
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To state the second conjecture, we need the following
result on ๐›พ-critical graphs.
Proposition 43. If ๐บ is a ๐›พ-critical graph, then ๐œ(๐บ) โฉฝ (ฮ”(๐บ)+
1)(๐›พ(๐บ)โˆ’1)+1; moreover, if the equality holds, then ๐บ is regular.
The upper bound in Proposition 43 is sharp in the
sense that equality holds for the infinite class of ๐›พ-critical
graphs ๐บ๐‘š,๐‘› defined in the beginning of this subsection. In
Proposition 43, the first result is due to Brigham et al. [24] in
1988; the second is due to Fulman et al. [27] in 1995.
For a ๐›พ-critical graph ๐บ with ๐›พ = 3, by Proposition 43,
๐œ(๐บ) โฉฝ 2ฮ”(๐บ) + 3.
Theorem 44 (Teschner [26], 1995). If ๐บ is a ๐›พ-critical graph
with ๐›พ = 3 and ๐œ(๐บ) = 2ฮ”(๐บ) + 3, then ๐‘(๐บ) โฉฝ ฮ” + 1.
We have not known yet whether the equality in
Theorem 44 holds or not. However, Samodivkin proposed the
following conjecture.
Conjecture 45 (Samodivkin [20], 2008). If ๐บ is a ๐›พ-critical
graph with (ฮ”(๐บ) + 1)(๐›พ โˆ’ 1) + 1 vertices, then ๐‘(๐บ) = ฮ”(๐บ) + 1.
In general, based on Theorem 41, Teschner proposed the
following conjecture.
Conjecture 46 (Teschner [26], 1995). If ๐บ is a ๐›พ-critical graph,
then ๐‘(๐บ) โฉฝ (3/2)ฮ”(๐บ) for ๐›พ โฉพ 4.
We conclude this subsection with some remarks. Graphs
which are minimal or critical with respect to a given property or parameter frequently play an important role in the
investigation of that property or parameter. Not only are such
graphs of considerable interest in their own right, but also a
knowledge of their structure often aids in the development
of the general theory. In particular, when investigating any
finite structure, a great number of results are proven by
induction. Consequently, it is desirable to learn as much as
possible about those graphs that are critical with respect to
a given property or parameter so as to aid and abet such
investigations.
9
In this subsection, we survey some results on the bondage
number for ๐›พ-critical graphs. Although these results are not
very perfect, they provides a feasible method to approach the
bondage number from different viewpoints. In particular, the
methods given in Teschner [26] worth further exploration
and development.
The following proposition is maybe useful for us to
further investigate the bondage number of a ๐›พ-critical graph.
Proposition 47 (Brigham et al. [24], 1988). If ๐บ has a
nonisolated vertex ๐‘ฅ such that the subgraph induced by ๐‘๐บ(๐‘ฅ)
is a complete graph, then ๐บ is not ๐›พ-critical.
This simple fact shows that any ๐›พ-critical graph contains
no vertices of degree one.
3.5. Bounds Implied by Domination. In the preceding subsection, we introduced some upper bounds on the bondage
numbers for ๐›พ-critical graphs by consideration of dominations. In this subsection, we introduce related results for a
general graph with given domination number.
Theorem 48 (Fink et al. [8], 1990). For any connected graph
๐บ of order ๐‘› (โฉพ 2),
(a) ๐‘(๐บ) โฉฝ ๐‘› โˆ’ 1;
(b) ๐‘(๐บ) โฉฝ ฮ”(๐บ) + 1 if ๐›พ(๐บ) = 2;
(c) ๐‘(๐บ) โฉฝ ๐‘› โˆ’ ๐›พ(๐บ) + 1.
While the upper bound of ๐‘›โˆ’1 on ๐‘(๐บ) is not particularly
good for many classes of graphs (e.g., trees and cycles), it is
an attainable bound. For example, if ๐บ is a complete ๐‘ก-partite
graph ๐พ๐‘ก (2) for ๐‘ก = ๐‘›/2 and ๐‘› โฉฝ 4 an even integer, then the
three bounds on ๐‘(๐บ) in Theorem 48 are sharp.
Teschner [26] consider a graph with ๐›พ(๐บ) = 1 or ๐›พ(๐บ) = 2.
The next result is almost trivial but useful.
Lemma 49. Let ๐บ be a graph with order ๐‘› and ๐›พ(๐บ) = 1, ๐‘ก the
number of vertices of degree ๐‘› โˆ’ 1. Then ๐‘(๐บ) = โŒˆ๐‘ก/2โŒ‰.
Since ๐‘ก โฉฝ ฮ”(๐บ) clearly, Lemma 49 yields the following
result immediately.
Theorem 50 (Teschner [26], 1995). ๐‘(๐บ) โฉฝ (1/2)ฮ” + 1 for any
graph ๐บ with ๐›พ(๐บ) = 1.
For a complete graph ๐พ2๐‘˜+1 , ๐‘(๐พ2๐‘˜+1 ) = ๐‘˜+1 = (1/2)ฮ”+1,
which shows that the upper bound given in Theorem 50 can
be attained. For a graph ๐บ with ๐›พ(๐บ) = 2, by Theorem 59 later,
the upper bound given in Theorem 48 (b) can be also attained
by a 2-critical graph (see Proposition 37).
3.6. Two Conjectures. In 1990, when Fink et al. [8] introduced
the concept of the bondage number; they proposed the
following conjecture.
Conjecture 51. ๐‘(๐บ) โฉฝ ฮ”(๐บ) + 1 for any nonempty graph ๐บ.
10
Although some results partially support Conjecture 51,
Teschner [28] in 1993 found a counterexample to this conjecture, the cartesian product ๐พ3 × ๐พ3 , which shows that
๐‘(๐บ) = ฮ”(๐บ) + 2.
If a graph ๐บ is a counterexample to Conjecture 51, it
must be a ๐›พ-critical graph by Theorem 13. That is why the
vertex domination-critical graphs are of special interest in the
literature.
Now, we return to Conjecture 51. Hartnell and Rall [17]
and Teschner [29], independently, proved that ๐‘(๐บ) can be
much greater than ฮ”(๐บ) by showing the following result.
Theorem 52 (Hartnell and Rall [17], 1994; Teschner [29].
1996). For an integer ๐‘› โฉพ 3, let ๐บ๐‘› be the cartesian product
๐พ๐‘› × ๐พ๐‘› . Then ๐‘(๐บ๐‘› ) = 3(๐‘› โˆ’ 1) = (3/2)ฮ”(๐บ๐‘› ).
This theorem shows that there exist no upper bounds
of the form ๐‘(๐บ) โฉฝ ฮ”(๐บ) + ๐‘ for any integer ๐‘. Teschner
[26] proved that ๐‘(๐บ) โฉฝ (3/2)ฮ”(๐บ) for any graph ๐บ with
๐›พ(๐บ) โฉฝ 2 (see Theorems 50 and 48) and for ๐›พ-critical graphs
with ๐›พ(๐บ) = 3 and proposed the following conjecture.
Conjecture 53 (Teschner [26], 1995). ๐‘(๐บ) โฉฝ (3/2)ฮ”(๐บ) for
any graph ๐บ.
We believe that this conjecture is true, but so far no much
work about it has been known.
4. Lower Bounds
Since the bondage number is defined as the smallest number
of edges whose removal results in an increase of domination
number, each constructive method that creates a concrete
bondage set leads to an upper bound on the bondage number.
For that reason, it is hard to find lower bounds. Nevertheless,
there are still a few lower bounds.
4.1. Bounds Implied by Subgraphs. The first lower bound
on the bondage number is gotten in terms of its spanning
subgraph.
Theorem 54 (Teschner [10], 1997). Let ๐ป be a spanning
subgraph of a nonempty graph ๐บ. If ๐›พ(๐ป) = ๐›พ(๐บ), then ๐‘(๐ป) โฉฝ
๐‘(๐บ).
By Theorem 1, ๐‘(๐ถ๐‘› ) = 3 if ๐‘› โ‰ก 1 (mod 3) and ๐‘(๐ถ๐‘› ) =
2, otherwise ๐‘(๐‘ƒ๐‘› ) = 2 if ๐‘› โ‰ก 1 (mod 3) and ๐‘(๐‘ƒ๐‘› ) = 1
otherwise. From these results and Theorem 54, we get the
following two corollaries.
Corollary 55. If ๐บ is hamiltonian with order ๐‘› โฉพ 3 and ๐›พ(๐บ) =
โŒˆ๐‘›/3โŒ‰, then ๐‘(๐บ) โฉพ 2 and in addition ๐‘(๐บ) โฉพ 3 if ๐‘› โ‰ก 1 (mod
3).
Corollary 56. If ๐บ with order ๐‘› โฉพ 2 has a hamiltonian path
and ๐›พ(๐บ) = โŒˆ๐‘›/3โŒ‰, then ๐‘(๐บ) โฉพ 2 if ๐‘› โ‰ก 1 (mod 3).
4.2. Other Bounds. The vertex covering number ๐›ฝ(๐บ) of ๐บ is
the minimum number of vertices that are incident with all
International Journal of Combinatorics
Figure 5: A ๐›พ-critical graph with ๐›ฝ = ๐›พ = 4, ๐›ฟ = 2, and ๐‘ = 3.
edges in ๐บ. If ๐บ has no isolated vertices, then ๐›พ(๐บ) โฉฝ ๐›ฝ(๐บ)
clearly. In [30], Volkmann gave a lot of graphs with ๐›ฝ = ๐›พ.
Theorem 57 (Teschner [10], 1997). Let ๐บ be a graph. If ๐›พ(๐บ) =
๐›ฝ(๐บ), then
(a) ๐‘(๐บ) โฉพ ๐›ฟ(๐บ);
(b) ๐‘(๐บ) โฉพ ๐›ฟ(๐บ) + 1 if ๐บ is a ๐›พ-critical graph.
The graph shown in Figure 5 shows that the bound given
in Theorem 57(b) is sharp. For the graph ๐บ, it is a ๐›พ-critical
graph and ๐›พ(๐บ) = ๐›ฝ(๐บ) = 4. By Theorem 57, we have ๐‘(๐บ) โฉพ
3. On the other hand, ๐‘(๐บ) โฉฝ 3 by Theorem 16. Thus, ๐‘(๐บ) =
3.
Proposition 58 (Sanchis [31], 1991). Let ๐บ be a graph ๐บ of
order ๐‘› (โฉพ 6) and size ๐œ€. If ๐บ has no isolated vertices and
).
3 โฉฝ ๐›พ(๐บ) โฉฝ ๐‘›/2, then ๐œ€ โฉฝ ( ๐‘›โˆ’๐›พ(๐บ)+1
2
Using the idea in the proof of Theorem 57, every upper
bound for ๐›พ(๐บ) can lead to a lower bound for ๐‘(๐บ). In
this way, Teschner [10] obtained another lower bound from
Proposition 58.
Theorem 59 (Teschner [10], 1997). Let ๐บ be a graph ๐บ of order
๐‘› (โฉพ 6) and size ๐œ€. If 2 โฉฝ ๐›พ(๐บ) โฉฝ ๐‘›/2 โˆ’ 1, then
)};
(a) ๐‘(๐บ) โฉพ min{๐›ฟ(๐บ), ๐œ€ โˆ’ ( ๐‘›โˆ’๐›พ(๐บ)
2
)} if ๐บ is a ๐›พ-critical
(b) ๐‘(๐บ) โฉพ min{๐›ฟ(๐บ) + 1, ๐œ€ โˆ’ ( ๐‘›โˆ’๐›พ(๐บ)
2
graph.
The lower bound in Theorem 59(b) is sharp for a class
of ๐›พ-critical graphs with domination number 2. Let ๐บ be the
graph obtained from complete graph ๐พ2๐‘ก (๐‘ก โฉพ 2) by removing
a perfect matching. By Proposition 37, ๐บ is a ๐›พ-critical graph
with ๐›พ(๐บ) = 2. Then ๐‘(๐บ) โฉพ ๐›ฟ(๐บ) + 1 = ฮ”(๐บ) + 1 by
Theorem 59 and ๐‘(๐บ) โฉฝ ฮ”(๐บ) + 1 by Theorem 38, and so
๐‘(๐บ) = ฮ”(๐บ) + 1 = 2๐‘ก โˆ’ 1.
As far as the author knows, there are no more lower
bounds in the present literature. In view of applications of
the bondage number, a network is vulnerable if its bondage
number is small while it is stable if its bondage number is
large. Therefore, better lower bounds let us learn better about
the stability of networks from this point of view. Thus, it is
of great significance to seek more lower bounds for various
classes of graphs.
International Journal of Combinatorics
11
5. Results on Graph Operations
Generally speaking, it is quite difficult to determine the exact
value of the bondage number for a given graph since it
strongly depends on the dominating number of the graph.
Thus, determining bondage numbers for some special graphs
is interesting if the dominating numbers of those graphs are
known or can be easily determined. In this section, we will
introduce some known results on bondage numbers for some
special classes of graphs obtained by some graph operations.
5.1. Cartesian Product Graphs. Let ๐บ1 = (๐‘‰1 , ๐ธ1 ) and ๐บ2 =
(๐‘‰2 , ๐ธ2 ) be two graphs. The Cartesian product of ๐บ1 and ๐บ2
is an undirected graph, denoted by ๐บ1 × ๐บ2 , where ๐‘‰(๐บ1 ×
๐บ2 ) = ๐‘‰1 × ๐‘‰2 ; two distinct vertices ๐‘ฅ1 ๐‘ฅ2 and ๐‘ฆ1 ๐‘ฆ2 , where
๐‘ฅ1 , ๐‘ฆ1 โˆˆ ๐‘‰(๐บ1 ) and ๐‘ฅ2 , ๐‘ฆ2 โˆˆ ๐‘‰(๐บ2 ), are linked by an edge in
๐บ1 × ๐บ2 if and only if either ๐‘ฅ1 = ๐‘ฆ1 and ๐‘ฅ2 ๐‘ฆ2 โˆˆ ๐ธ(๐บ2 ), or
๐‘ฅ2 = ๐‘ฆ2 and ๐‘ฅ1 ๐‘ฆ1 โˆˆ ๐ธ(๐บ1 ). The Cartesian product is a very
effective method for constructing a larger graph from several
specified small graphs.
Theorem 60 (Dunbar et al. [21], 1998). For ๐‘› โฉพ 3,
2
{
{
๐‘ (๐ถ๐‘› × ๐พ2 ) = {3
{
{4
๐‘–๐‘“ ๐‘› โ‰ก 0 ๐‘œ๐‘Ÿ 1 (mod 4) ,
๐‘–๐‘“ ๐‘› โ‰ก 3 (mod 4) ,
๐‘–๐‘“ ๐‘› โ‰ก 2 (mod 4) .
(16)
Theorem 61 (Sohn et al. [34], 2007). For ๐‘› โฉพ 4,
๐‘–๐‘“ ๐‘› โ‰ก 0 (mod 4) ,
๐‘–๐‘“ ๐‘› โ‰ก 1 ๐‘œ๐‘Ÿ 2 (mod 4) ,
๐‘–๐‘“ ๐‘› โ‰ก 3 (mod 4) .
(17)
Theorem 62 (Kang et al. [35], 2005). ๐‘(๐ถ๐‘› ×๐ถ4 ) = 4 for ๐‘› โฉพ 4.
For larger ๐‘š and ๐‘›, Huang and Xu [36] obtained the
following result; see Theorem 197 for more details.
Theorem 63 (Huang and Xu [36], 2008). ๐‘(๐ถ5๐‘– × ๐ถ5๐‘— ) = 3 for
any positive integers ๐‘– and ๐‘—.
Xiang et al. [37] determined that for ๐‘› โฉพ 5,
=3
{
{
๐‘ (๐ถ๐‘› × ๐ถ5 ) {= 4
{
{โฉฝ 7
if ๐‘› โ‰ก 0 or 1 (mod 5) ,
if ๐‘› โ‰ก 2 or 4 (mod 5) ,
if ๐‘› โ‰ก 3 (mod 5) .
Theorem 64 (Hu and Xu [40], 2012). For ๐‘› โฉพ 4,
1,
๐‘ (๐‘ƒ๐‘› × ๐‘ƒ2 ) = {
2,
๐‘–๐‘“ ๐‘› โ‰ก 1 (mod 2)
๐‘–๐‘“ ๐‘› โ‰ก 0 (mod 2) ;
1,
๐‘ (๐‘ƒ๐‘› × ๐‘ƒ3 ) = {
2,
๐‘–๐‘“ ๐‘› โ‰ก 1 ๐‘œ๐‘Ÿ 2 (mod 4)
๐‘–๐‘“ ๐‘› โ‰ก 0 ๐‘œ๐‘Ÿ 3 (mod 4) ,
(19)
๐‘ (๐‘ƒ๐‘› × ๐‘ƒ4 ) = 1 ๐‘“๐‘œ๐‘Ÿ ๐‘› โฉพ 14.
From the proof of Theorem 64, we find that if ๐‘ƒ๐‘› =
{0, 1, . . . , ๐‘› โˆ’ 1} and ๐‘ƒ๐‘š = {0, 1, . . . , ๐‘š โˆ’ 1}, then the removal
of the vertex (0, 0) in ๐‘ƒ๐‘› × ๐‘ƒ๐‘š does not change the domination
number. If ๐‘š increases, the effect of (0, 0) for the domination
number will be smaller and smaller from the probability.
Therefore, we expect it is possible that ๐›พ(๐‘ƒ๐‘› × ๐‘ƒ๐‘š โˆ’ (0, 0)) =
๐›พ(๐‘ƒ๐‘› × ๐‘ƒ๐‘š ) for ๐‘š โฉพ 5 and give the following conjecture.
Conjecture 65. ๐‘(๐‘ƒ๐‘› × ๐‘ƒ๐‘š ) โฉฝ 2 for ๐‘š โฉพ 5 and ๐‘› โฉพ 4.
For the Cartesian product ๐ถ๐‘› × ๐ถ๐‘š of two cycles ๐ถ๐‘› and
๐ถ๐‘š , where ๐‘› โฉพ 4 and ๐‘š โฉพ 3, KlavzฬŒar and Seifter [32]
determined ๐›พ(๐ถ๐‘› × ๐ถ๐‘š ) for some ๐‘› and ๐‘š. For example,
๐›พ(๐ถ๐‘› × ๐ถ4 ) = ๐‘› for ๐‘› โฉพ 3 and ๐›พ(๐ถ๐‘› × ๐ถ3 ) = ๐‘› โˆ’ โŒŠ๐‘›/4โŒ‹ for ๐‘› โฉพ 4.
Kim [33] determined ๐‘(๐ถ3 × ๐ถ3 ) = 6 and ๐‘(๐ถ4 × ๐ถ4 ) = 4. For
a general ๐‘› โฉพ 4, the exact values of the bondage numbers of
๐ถ๐‘› × ๐ถ3 and ๐ถ๐‘› × ๐ถ4 are determined as follows.
2
{
{
๐‘ (๐ถ๐‘› × ๐ถ3 ) = {4
{
{5
for ๐‘› โฉพ 4 and 2 โฉฝ ๐‘š โฉฝ 4 is determined as follows (Li [39] also
determined ๐‘(๐‘ƒ๐‘› × ๐‘ƒ2 )).
(18)
For the Cartesian product ๐‘ƒ๐‘› × ๐‘ƒ๐‘š of two paths ๐‘ƒ๐‘› and ๐‘ƒ๐‘š ,
Jacobson and Kinch [38] determined ๐›พ(๐‘ƒ๐‘› × ๐‘ƒ2 ) = โŒˆ(๐‘› + 1)/2โŒ‰,
๐›พ(๐‘ƒ๐‘› × ๐‘ƒ3 ) = ๐‘› โˆ’ โŒŠ(๐‘› โˆ’ 1)/4โŒ‹, and ๐›พ(๐‘ƒ๐‘› × ๐‘ƒ4 ) = ๐‘› + 1 if ๐‘› =
1, 2, 3, 5, 6, 9, and ๐‘› otherwise. The bondage number of ๐‘ƒ๐‘› ×๐‘ƒ๐‘š
5.2. Block Graphs and Cactus Graphs. In this subsection, we
introduce some results for corona graphs, block graphs, and
cactus graphs.
The corona ๐บ1 โˆ˜ ๐บ2 , proposed by Frucht and Harary [41],
is the graph formed from a copy of ๐บ1 and ๐œ(๐บ1 ) copies of
๐บ2 by joining the ๐‘–th vertex of ๐บ1 to the ๐‘–th copy of ๐บ2 . In
particular, we are concerned with the corona ๐ปโˆ˜๐พ1 , the graph
formed by adding a new vertex V๐‘– , and a new edge ๐‘ข๐‘– V๐‘– for
every vertex ๐‘ข๐‘– in ๐ป.
Theorem 66 (Carlson and Develin [42], 2006). ๐‘(๐ป โˆ˜ ๐พ1 ) =
๐›ฟ(๐ป) + 1.
This is a very important result, which is often used to
construct a graph with required bondage number. In other
words, this result implies that for any given positive integer ๐‘
there is a graph ๐บ such that ๐‘(๐บ) = ๐‘.
A block graph is a graph whose blocks are complete
graphs. Each block in a cactus graph is either a cycle or a ๐พ2 .
If each block of a graph is either a complete graph or a cycle,
then we call this graph a block-cactus graph.
Theorem 67 (Teschner [43], 1997). ๐‘(๐บ) โ‰ค ฮ”(๐บ) for any block
graph ๐บ.
Teschner [43] characterized all block graphs with ๐‘(๐บ) =
ฮ”(๐บ). In the same paper, Teschner found that ๐›พ-critical
graphs are instrumental in determining bounds for the
bondage number of cactus and block graphs and obtained the
following result.
Theorem 68 (Teschner [43], 1997). ๐‘(๐บ)
nontrivial cactus graph ๐บ.
โฉฝ
3 for any
This bound can be achieved by ๐ป โˆ˜ ๐พ1 where ๐ป is
a nontrivial cactus graph without vertices of degree one
12
International Journal of Combinatorics
Problem 2. Characterize all cactus graphs with bondage
number 3.
๐‘ฅ and joining it to an vertex ๐‘ฆ of degree one in ๐ป โˆ˜ ๐พ1 . Then
๐บ/๐‘ฅ๐‘ฆ = ๐ป โˆ˜ ๐พ1 , ๐›พ(๐บ) = ๐›พ(๐บ/๐‘ฅ๐‘ฆ), and ๐‘๐บ(๐‘ฅ) โˆฉ ๐‘๐บ(๐‘ฆ) = 0.
But ๐‘(๐บ) = 1 since ๐›พ(๐บ โˆ’ ๐‘ฅ๐‘ฆ) = ๐›พ(๐บ/๐‘ฅ๐‘ฆ) + 1, and ๐‘(๐บ/๐‘ฅ๐‘ฆ) =
๐›ฟ(๐ป) + 1 by Theorem 66. The gap between ๐‘(๐บ) and ๐‘(๐บ/๐‘ฅ๐‘ฆ)
is ๐›ฟ(๐ป).
Some upper bounds for block-cactus graphs were also
obtained.
6. Results on Planar Graphs
by Theorem 66. In 1998, Dunbar et al. [21] proposed the
following problem.
Theorem 69 (Dunbar et al. [21], 1998). Let ๐บ be a connected
block-cactus graph with at least two blocks. Then ๐‘(๐บ) โฉฝ ฮ”(๐บ).
Theorem 70 (Dunbar et al. [21], 1998). Let ๐บ be a connected
block-cactus graph which is neither a cactus graph nor a block
graph. Then ๐‘(๐บ) โฉฝ ฮ”(๐บ) โˆ’ 1.
5.3. Edge-Deletion and Edge-Contraction. In order to obtain
further results of the bondage number, we may consider how
the bondage number changes under some operations of a
graph ๐บ, which remain the domination number unchanged
or preserve some property of ๐บ, such as planarity. Two
simple operations satisfying these requirements are the edgedeletion and the edge-contraction.
Theorem 71 (Huang and Xu [44], 2012). Let ๐บ be any graph
and ๐‘’ โˆˆ ๐ธ(๐บ). Then ๐‘(๐บ โˆ’ ๐‘’) โฉพ ๐‘(๐บ) โˆ’ 1. Moreover, ๐‘(๐บ โˆ’ ๐‘’) โฉฝ
๐‘(๐บ) if ๐›พ(๐บ โˆ’ ๐‘’) = ๐›พ(๐บ).
From Theorem 1, ๐‘(๐ถ๐‘› โˆ’ ๐‘’) = ๐‘(๐‘ƒ๐‘› ) = ๐‘(๐ถ๐‘› ) โˆ’ 1 for any ๐‘’
in ๐ถ๐‘› , which shows that the lower bound on ๐‘(๐บ โˆ’ ๐‘’) is sharp.
The upper bound can reach for any graph ๐บ with ๐‘(๐บ) โฉพ 2.
Next, we consider the effect of the edge-contraction on
the bondage number. Given a graph ๐บ, the contraction of ๐บ
by the edge ๐‘’ = ๐‘ฅ๐‘ฆ, denoted by ๐บ/๐‘ฅ๐‘ฆ, is the graph obtained
from ๐บ by contracting two vertices ๐‘ฅ and ๐‘ฆ to a new vertex
and then deleting all multiedges. It is easy to observe that for
any graph ๐บ, ๐›พ(๐บ) โˆ’ 1 โฉฝ ๐›พ(๐บ/๐‘ฅ๐‘ฆ) โฉฝ ๐›พ(๐บ) for any edge ๐‘ฅ๐‘ฆ of
๐บ.
Theorem 72 (Huang and Xu [44], 2012). Let ๐บ be any graph
and ๐‘ฅ๐‘ฆ be any edge in ๐บ. If ๐‘๐บ(๐‘ฅ) โˆฉ ๐‘๐บ(๐‘ฆ) = 0 and ๐›พ(๐บ/๐‘ฅ๐‘ฆ) =
๐›พ(๐บ), then ๐‘(๐บ/๐‘ฅ๐‘ฆ) โฉพ ๐‘(๐บ).
We can use examples ๐ถ๐‘› and ๐พ๐‘› to show that the
conditions of Theorem 72 are necessary and the lower bound
in Theorem 72 is tight. Clearly, ๐›พ(๐ถ๐‘› ) = โŒˆ๐‘›/3โŒ‰ and ๐›พ(๐พ๐‘› ) = 1;
for any edge ๐‘ฅ๐‘ฆ, ๐ถ๐‘› /๐‘ฅ๐‘ฆ = ๐ถ๐‘›โˆ’1 and ๐พ๐‘› /๐‘ฅ๐‘ฆ = ๐พ๐‘›โˆ’1 . By
Theorem 1, if ๐‘› โ‰ก 1 (mod 3), then ๐›พ(๐ถ๐‘› /๐‘ฅ๐‘ฆ) < ๐›พ(๐บ) and
๐‘(๐ถ๐‘› /๐‘ฅ๐‘ฆ) = 2 < 3 = ๐‘(๐ถ๐‘› ). Thus, the result in Theorem 72
is generally invalid without the hypothesis ๐›พ(๐บ/๐‘ฅ๐‘ฆ) = ๐›พ(๐บ).
Furthermore, the condition ๐‘๐บ(๐‘ฅ) โˆฉ ๐‘๐บ(๐‘ฆ) = 0 cannot be
omitted even if ๐›พ(๐บ/๐‘ฅ๐‘ฆ) = ๐›พ(๐บ), since for odd ๐‘›, ๐‘(๐พ๐‘› /๐‘ฅ๐‘ฆ) =
โŒˆ(๐‘› โˆ’ 1)/2โŒ‰ < โŒˆ๐‘›/2โŒ‰ = ๐‘(๐พ๐‘› ), by Theorem 1.
On the other hand, ๐‘(๐ถ๐‘› /๐‘ฅ๐‘ฆ) = ๐‘(๐ถ๐‘› ) = 2 if ๐‘› โ‰ก 0 (mod
3) and ๐‘(๐พ๐‘› /๐‘ฅ๐‘ฆ) = ๐‘(๐พ๐‘› ) if ๐‘› is even, which shows that the
equality in ๐‘(๐บ/๐‘ฅ๐‘ฆ) โฉพ ๐‘(๐บ) may hold. Thus, the bound in
Theorem 72 is tight. However, provided all the conditions,
๐‘(๐บ/๐‘ฅ๐‘ฆ) can be arbitrarily larger than ๐‘(๐บ). Given a graph ๐ป,
let ๐บ be the graph formed from ๐ปโˆ˜๐พ1 by adding a new vertex
From Section 2, we have seen that the bondage number
for a tree has been completely solved. Moreover, a linear
time algorithm to compute the bondage number of a tree is
designed by Hartnell et al. [15]. It is quite natural to consider
the bondage number for a planar graph. In this section, we
state some results and problems on the bondage number for
a planar graph.
6.1. Conjecture on Bondage Number. As mentioned in
Section 3, the bondage number can be much larger than the
maximum degree. But for a planar graph ๐บ, the bondage
number ๐‘(๐บ) cannot exceed ฮ”(๐บ) too much. It is clear that
๐‘(๐บ) โฉฝ ฮ”(๐บ) + 4 by Corollary 15 since ๐›ฟ(๐บ) โฉฝ 5 for any
planar graph ๐บ. In 1998, Dunbar et al. [21] posed the following
conjecture.
Conjecture 73. If ๐บ is a planar graph, then ๐‘(๐บ) โฉฝ ฮ”(๐บ) + 1.
Because of its attraction, it immediately became the focus
of attention when this conjecture is proposed. In fact, the
main aim concerning the research of the bondage number for
a planar graph is focused on this conjecture.
It has been mentioned in Theorems 1 and 60 that
๐‘(๐ถ3๐‘˜+1 ) = 3 = ฮ” + 1, and ๐‘(๐ถ4๐‘˜+2 × ๐พ2 ) = 4 = ฮ” + 1. It
is known that ๐‘(๐พ6 โˆ’ ๐‘€) = 5 = ฮ” + 1, where ๐‘€ is a perfect
matching of the complete graph ๐พ6 . These examples show that
if Conjecture 73 is true, then the upper bound is sharp for
2 โฉฝ ฮ” โฉฝ 4.
Here, we note that it is sufficient to prove this conjecture
for connected planar graphs, since the bondage number of a
disconnected graph is simply the minimum of the bondage
numbers of its components.
The first paper attacking this conjecture is due to Kang
and Yuan [45], which confirmed the conjecture for every
connected planar graph ๐บ with ฮ” โฉพ 7. The proofs are mainly
based on Theorems 16 and 18.
Theorem 74 (Kang and Yuan [45], 2000). If ๐บ is a connected
planar graph, then ๐‘(๐บ) โฉฝ min{8, ฮ”(๐บ) + 2}.
Obviously, in view of Theorem 74, Conjecture 73 is true
for any connected planar graph with ฮ” โฉพ 7.
6.2. Bounds Implied by Degree Conditions. As we have seen
from Theorem 74, to attack Conjecture 73, we only need to
consider connected planar graphs with maximum ฮ” โฉฝ 6.
Thus, studying the bondage number of planar graphs by
degree conditions is of significance. The first result on bounds
implied by degree conditions is obtained by Kang and Yuan
[45].
International Journal of Combinatorics
13
Theorem 75 (Kang and Yuan [45], 2000). If ๐บ is a connected
planar graph without vertices of degree 5, then ๐‘(๐บ) โฉฝ 7.
Fischermann et al. [46] generalized Theorem 75 as follows.
Theorem 76 (Fischermann et al. [46], 2003). Let ๐บ be a
connected planar graph and ๐‘‹ the set of vertices of degree 5
which have distance at least 3 to vertices of degrees 1, 2, and
3. If all vertices in ๐‘‹ not adjacent with vertices of degree 4
are independent and not adjacent to vertices of degree 6, then
๐‘(๐บ) โฉฝ 7.
The first result in Theorem 79 also implies that if ๐บ is
a connected planar graph with no cycles of length 3, then
๐‘(๐บ) โฉฝ 6. Hou et al. [47] improved this result as follows.
Theorem 81 (Hou et al. [47], 2011). For any connected planar
graph ๐บ, if ๐บ contains no cycles of length ๐‘– (4 โฉฝ ๐‘– โฉฝ 6), then
๐‘(๐บ) โฉฝ 6.
Since ๐‘(๐ถ3๐‘˜+1 ) = 3 for ๐‘˜ โฉพ 3 (see Theorem 1), the
last bound in Theorem 79 is tight. Whether other bounds in
Theorem 79 are tight remains open. In 2003, Fischermann et
al. [46] posed the following conjecture.
Clearly, if ๐บ has no vertices of degree 5, then ๐‘‹ = 0. Thus,
Theorem 76 yields Theorem 75.
Use ๐‘›๐‘– to denote the number of vertices of degree ๐‘– in ๐บ
for each ๐‘– = 1, 2, . . . , ฮ”(๐บ). Using Theorem 18, Fischermann
et al. obtained the following two theorems.
Conjecture 82. For any connected planar graph ๐บ, ๐‘(๐บ) โฉฝ 7,
and
Theorem 77 (Fischermann et al. [46], 2003). For any connected planar graph ๐บ,
We conclude this subsection with a question on bondage
numbers of planar graphs.
(1) ๐‘(๐บ) โฉฝ 7 if ๐‘›5 < 2๐‘›2 + 3๐‘›3 + 2๐‘›4 + 12;
(2) ๐‘(๐บ) โฉฝ 6 if ๐บ contains no vertices of degrees 4 and 5.
Theorem 78 (Fischermann et al. [46], 2003). For any connected planar graph ๐บ, ๐‘(๐บ) โฉฝ ฮ”(๐บ) + 1 if
(1) ฮ”(๐บ) = 6 and every edge ๐‘’ = ๐‘ฅ๐‘ฆ with ๐‘‘๐บ(๐‘ฅ) = 5 and
๐‘‘๐บ(๐‘ฆ) = 6 is contained in at most one triangle;
(2) ฮ”(๐บ) = 5 and no triangle contains an edge ๐‘’ = ๐‘ฅ๐‘ฆ with
๐‘‘๐บ(๐‘ฅ) = 5 and 4 โฉฝ ๐‘‘๐บ(๐‘ฆ) โฉฝ 5.
6.3. Bounds Implied by Girth Conditions. The girth ๐‘”(๐บ) of a
graph ๐บ is the length of the shortest cycle in ๐บ. If ๐บ has no
cycles, we define ๐‘”(๐บ) = โˆž.
Combining Theorem 74 with Theorem 78, we find that
if a planar graph contains no triangles and has maximum
degree ฮ” โฉพ 5, then Conjecture 73 holds. This fact motivated
Fischermann et al.โ€™s [46] attempt to attack Conjecture 73 by
girth restraints.
Theorem 79 (Fischermann et al. [46], 2003). For any connected planar graph ๐บ,
6
{
{
{
{5
๐‘ (๐บ) โฉฝ {
{
4
{
{
{3
๐‘–๐‘“
๐‘–๐‘“
๐‘–๐‘“
๐‘–๐‘“
๐‘” (๐บ) โฉพ 4,
๐‘” (๐บ) โฉพ 5,
๐‘” (๐บ) โฉพ 6,
๐‘” (๐บ) โฉพ 8.
(20)
The first result in Theorem 79 shows that Conjecture 73
is valid for all connected planar graphs with ๐‘”(๐บ) โฉพ 4 and
ฮ”(๐บ) โฉพ 5. It is easy to verify that the second result in
Theorem 79 implies that Conjecture 73 is valid for all not
3-regular graphs of girth ๐‘”(๐บ) โฉพ 5, which is stated in the
following corollary.
Corollary 80. For any connected planar graph ๐บ, if ๐บ is not
3-regular and ๐‘”(๐บ) โฉพ 5, then ๐‘(๐บ) โฉฝ ฮ”(๐บ) + 1.
5,
๐‘ (๐บ) โฉฝ {
4,
๐‘–๐‘“ ๐‘” (๐บ) โฉพ 4,
๐‘–๐‘“ ๐‘” (๐บ) โฉพ 5.
(21)
Question 1 (Fischermann et al. [46], 2003). Is there a planar
graph ๐บ with 6 โฉฝ ๐‘(๐บ) โฉฝ 8?
In 2006, Carlson and Develin [42] showed that the corona
๐บ = ๐ป โˆ˜ ๐พ1 for a planar graph ๐ป with ๐›ฟ(๐ป) = 5 has the
bondage number ๐‘(๐บ) = ๐›ฟ(๐ป) + 1 = 6 (see Theorem 66).
Since the minimum degree of planar graphs is at most 5,
then ๐‘(๐บ) can attach 6. If we take ๐ป as the graph of the
icosahedron, then ๐บ = ๐ป โˆ˜ ๐พ1 is such an example. The
question for the existence of planar graphs with bondage
number 7 or 8 remains open.
6.4. Comments on the Conjectures. Conjecture 73 is true for
all connected planar graphs with minimum degree ๐›ฟ โฉฝ 2 by
Theorem 16, or maximum degree ฮ” โฉพ 7 by Theorem 74, or
not ๐›พ-critical planar graphs by Theorem 13. Thus, to attack
Conjecture 73, we only need to consider connected critical
planar graphs with degree-restriction 3 โฉฝ ๐›ฟ โฉฝ ฮ” โฉฝ 6.
Recalling and anatomizing the proofs of all results mentioned in the preceding subsections on the bondage number
for connected planar graphs, we find that the proofs of these
results strongly depend upon Theorem 16 or Theorem 18. In
other words, a basic way used in the proofs is to find two
vertices ๐‘ฅ and ๐‘ฆ with distance at most two in a considered
planar graph ๐บ such that
๐‘‘๐บ (๐‘ฅ) + ๐‘‘๐บ (๐‘ฆ)
๓ต„จ
๓ต„จ
or ๐‘‘๐บ (๐‘ฅ) + ๐‘‘๐บ (๐‘ฆ) โˆ’ ๓ต„จ๓ต„จ๓ต„จ๐‘๐บ (๐‘ฅ) โˆฉ ๐‘๐บ (๐‘ฆ)๓ต„จ๓ต„จ๓ต„จ ,
(22)
which bounds ๐‘(๐บ), is as small as possible. Very recently,
Huang and Xu [44] have considered the following parameter
๐ต (๐บ) = min {{๐‘‘๐‘ฅ๐‘ฆ : 1 โฉฝ ๐‘‘๐บ (๐‘ฅ, ๐‘ฆ) โฉฝ 2}
๐‘ฅ,๐‘ฆโˆˆ๐‘‰(๐บ)
(23)
โˆช {๐‘‘๐‘ฅ๐‘ฆ โˆ’ ๐‘›๐‘ฅ๐‘ฆ : ๐‘‘ (๐‘ฅ, ๐‘ฆ) = 1}} ,
where ๐‘‘๐‘ฅ๐‘ฆ = ๐‘‘๐บ(๐‘ฅ) + ๐‘‘๐บ(๐‘ฆ) โˆ’ 1 and ๐‘›๐‘ฅ๐‘ฆ = |๐‘๐บ(๐‘ฅ) โˆฉ ๐‘๐บ(๐‘ฆ)|.
14
International Journal of Combinatorics
(24)
7.1. General Methods. Use ๐‘›๐‘– to denote the number of vertices
of degree ๐‘– in ๐บ for ๐‘– = 1, 2, . . . , ฮ”(๐บ). Huang and Xu obtained
the following results.
The proofs given in Theorems 74 and 79 indeed imply the
following stronger results.
Theorem 86 (Huang and Xu [48], 2007). For any connected
graph ๐บ,
It follows from Theorems 16 and 18 that
๐‘ (๐บ) โฉฝ ๐ต (๐บ) .
Theorem 83. If ๐บ is a connected planar graph, then
min {8, ฮ” (๐บ) + 2}
{
{
{
{
{ 6 ๐‘–๐‘“ ๐‘” (๐บ) โฉพ 4,
๐ต (๐บ) โฉฝ { 5 ๐‘–๐‘“ ๐‘” (๐บ) โฉพ 5,
{ 4 ๐‘–๐‘“ ๐‘” (๐บ) โฉพ 6,
{
{
{
{ 3 ๐‘–๐‘“ ๐‘” (๐บ) โฉพ 8.
(25)
Thus, using Theorem 16 or Theorem 18, if we can prove
Conjectures 73 and 82, then we can prove the following
statement.
Statement. If ๐บ is a connected planar graph, then
ฮ” (๐บ) + 1,
{
{
{
{7,
๐ต (๐บ) โฉฝ {
{
5,
if ๐‘” (๐บ) โฉพ 4,
{
{
4,
if ๐‘” (๐บ) โฉพ 5.
{
(26)
It follows from (24) that Statement implies Conjectures 73
and 82. However, Huang and Xu [44] gave examples to show
that none of conclusions in Statement is true. As a result, they
stated the following conclusion.
Theorem 84 (Huang and Xu [44], 2012). It is not possible to
prove Conjectures 73 and 82, if they are right, using Theorems
16 and 18.
Therefore, a new method is needed to prove or disprove
these conjectures. At the present time, one cannot prove these
conjectures, and may consider to disprove them. If one of
these conjectures is invalid, then there exists a minimum
counterexample ๐บ with respect to ๐œ(๐บ) + ๐œ€(๐บ). In order to
obtain a minimum counterexample, one may consider two
simple operations satisfying these requirements, the edgedeletion and the edge-contraction, which decrease ๐œ(๐บ)+๐œ€(๐บ)
and preserve planarity. However, applying Theorems 71 and
72, Huang and Xu [44] presented the following results.
Theorem 85 (Huang and Xu [44], 2012). It is not possible to
construct minimum counterexamples to Conjectures 73 and 82
by the operation of an edge-deletion or an edge-contraction.
7. Results on Crossing Number Restraints
It is quite natural to generalize the known results on the
bondage number for planar graphs to for more general graphs
in terms of graph-theoretical parameters. In this section, we
consider graphs with crossing number restraints.
The crossing number cr(๐บ) of ๐บ is the smallest number
of pairwise intersections of its edges when ๐บ is drawn in the
plane. If cr(๐บ) = 0, then ๐บ is a planar graph.
{
6
{
{
{
{
{
{
{
{
{
{5
๐‘ (๐บ) โฉฝ {
{
{
{
4
{
{
{
{
{
{
{3
{
1
๐‘–๐‘“ ๐‘” (๐บ) โฉพ 4 ๐‘Ž๐‘›๐‘‘ 2cr (๐บ) < ๐‘Ž1 ;
2
1
๐‘–๐‘“ ๐‘” (๐บ) โฉพ 5 ๐‘Ž๐‘›๐‘‘ 6cr (๐บ) < ๐‘Ž2 ;
6
1
๐‘–๐‘“ ๐‘” (๐บ) โฉพ 6 ๐‘Ž๐‘›๐‘‘ 4cr (๐บ) < ๐‘Ž3 ;
4
1
๐‘–๐‘“ ๐‘” (๐บ) โฉพ 8 ๐‘Ž๐‘›๐‘‘ 6cr (๐บ) < ๐‘Ž4 ,
6
(27)
where ๐‘Ž1 = ๐‘›1 + 2๐‘›2 + 2๐‘›3 + โˆ‘ฮ”๐‘–=8 (๐‘– โˆ’ 7)๐‘›๐‘– + 8; ๐‘Ž2 = 3๐‘›1 + 6๐‘›2 +
5๐‘›3 + โˆ‘ฮ”๐‘–=7 (3๐‘– โˆ’ 18)๐‘›๐‘– + 20; ๐‘Ž3 = ๐‘›1 + 2๐‘›2 + โˆ‘ฮ”๐‘–=6 (2๐‘– โˆ’ 10)๐‘›๐‘– + 12;
and ๐‘Ž4 = โˆ‘ฮ”๐‘–=5 (3๐‘– โˆ’ 12)๐‘›๐‘– + 16.
Simplifying the conditions in Theorem 86, we obtain the
following corollaries immediately.
Corollary 87 (Huang and Xu [48], 2007). For any connected
graph ๐บ,
6,
{
{
{
{
{
{5,
{
{
๐‘ (๐บ) โฉฝ {
{4,
{
{
{
{
{
{
{3,
๐‘–๐‘“ ๐‘” (๐บ) โฉพ 4 ๐‘Ž๐‘›๐‘‘ cr (๐บ) โฉฝ 3,
๐‘–๐‘“ ๐‘” (๐บ) โฉพ 5 ๐‘Ž๐‘›๐‘‘ cr (๐บ) โฉฝ 4,
๐‘–๐‘“ ๐‘” (๐บ) โฉพ 6 ๐‘Ž๐‘›๐‘‘ cr (๐บ) โฉฝ 2,
(28)
๐‘–๐‘“ ๐‘” (๐บ) โฉพ 8 ๐‘Ž๐‘›๐‘‘ cr (๐บ) โฉฝ 2.
Corollary 88 (Huang and Xu [48], 2007). For any connected
graph ๐บ,
(a) ๐‘(๐บ) โฉฝ 6 if ๐บ is not 4-regular, cr(๐บ) = 4 and ๐‘”(๐บ) โฉพ 4;
(b) ๐‘(๐บ) โฉฝ ฮ”(๐บ) + 1 if ๐บ is not 3-regular, cr(๐บ) โฉฝ 4 and
๐‘”(๐บ) โฉพ 5;
(c) ๐‘(๐บ) โฉฝ 4 if ๐บ is not 3-regular, cr(๐บ) = 3 and ๐‘”(๐บ) โฉพ 6;
(d) ๐‘(๐บ) โฉฝ 3 if cr(๐บ) = 3, ๐‘”(๐บ) โฉพ 8 and ฮ”(๐บ) โฉพ 5.
These corollaries generalize some known results for planar graphs. For example, Corollary 87 contains Theorem 79;
Corollary 88 (b) contains Corollary 80.
Theorem 89 (Huang and Xu [48], 2007). For any connected
graph ๐บ, if cr(๐บ) โฉฝ ๐‘›3 (๐บ) + ๐‘›4 (๐บ) + 3, then ๐‘(๐บ) โฉฝ 8.
Corollary 90 (Huang and Xu [48], 2007). For any connected
graph ๐บ, if cr(๐บ) โฉฝ 3, then ๐‘(๐บ) โฉฝ 8.
Perhaps being unaware of this result, in 2010, Ma et al.
[49] proved that ๐‘(๐บ) โฉฝ 12 for any graph ๐บ with cr(๐บ) = 1.
Theorem 91 (Huang and Xu [48], 2007). Let ๐บ be a connected
graph, and ๐ผ = {๐‘ฅ โˆˆ ๐‘‰(๐บ) : ๐‘‘๐บ(๐‘ฅ) = 5, ๐‘‘๐บ(๐‘ฅ, ๐‘ฆ) โฉพ 3 if
International Journal of Combinatorics
15
๐‘‘๐บ(๐‘ฆ) โฉฝ 3, and ๐‘‘๐บ(๐‘ฆ) =ฬธ 4 for every ๐‘ฆ โˆˆ ๐‘๐บ(๐‘ฅ)}. If ๐ผ is
independent, no vertices adjacent to vertices of degree 6 and
cr (๐บ) < max {
5๐‘›3 + |๐ผ| โˆ’ 2๐‘›4 + 28 7๐‘›3 + 40
,
},
11
16
(29)
then ๐‘(๐บ) โฉฝ 7.
Corollary 92 (Huang and Xu [48], 2007). Let ๐บ be a
connected graph with cr(๐บ) โฉฝ 2. If ๐ผ = {๐‘ฅ โˆˆ ๐‘‰(๐บ) : ๐‘‘๐บ(๐‘ฅ) =
5, ๐‘‘๐บ(๐‘ฆ, ๐‘ฅ) โฉพ 3 if ๐‘‘๐บ(๐‘ฆ) โฉฝ 3 and ๐‘‘๐บ(๐‘ฆ) =ฬธ 4 for every ๐‘ฆ โˆˆ
๐‘๐บ(๐‘ฅ)} is independent, and no vertices adjacent to vertices of
degree 6, then ๐‘(๐บ) โฉฝ 7.
Theorem 93 (Huang and Xu [48], 2007). Let ๐บ be a connected
graph. If ๐บ satisfies
(a) 5cr(๐บ) + ๐‘›5 < 2๐‘›2 + 3๐‘›3 + 2๐‘›4 + 12 or
(b) 7cr(๐บ) + 2๐‘›5 < 3๐‘›2 + 4๐‘›4 + 24,
then ๐‘(๐บ) โฉฝ 7.
Proposition 94 (Huang and Xu [48], 2007). Let ๐บ be a
connected graph with no vertices of degrees four and five. If
cr(๐บ) โฉฝ 2, then ๐‘(๐บ) โฉฝ 6.
Theorem 95 (Huang and Xu [48], 2007). If ๐บ is a connected
graph with cr(๐บ) โฉฝ 4 and not 4-regular when cr(๐บ) = 4, then
๐‘(๐บ) โฉฝ ฮ”(๐บ) + 2.
The above results generalize some known results for
planar graphs. For example, Corollary 90 and Theorem 95
contain Theorem 74; the first condition in Theorem 93 contains the second condition in Theorem 77.
From Corollary 88 and Theorem 95, we suggest the following questions.
Question 2. Is there a
(a) 4-regular graph ๐บ with cr(๐บ) = 4 such that ๐‘(๐บ) โฉพ 7?
(b) 3-regular graph ๐บ with cr โฉฝ 4 and ๐‘”(๐บ) โฉพ 5 such that
๐‘(๐บ) = 5?
(c) 3-regular graph ๐บ with cr = 3 and ๐‘”(๐บ) = 6 or 7 such
that ๐‘(๐บ) = 5?
7.2. Carlson-Develin Methods. In this subsection, we introduce an elegant method presented by Carlson and Develin
[42] to obtain some upper bounds for the bondage number
of a graph.
Suppose that ๐บ is a connected graph. We say that ๐บ has
genus ๐œŒ if ๐บ can be embedded in a surface ๐‘† with ๐œŒ handles
such that edges are pairwise disjoint except possibly for endฬƒ be an embedding of ๐บ in surface ๐‘†, and let ๐œ™(๐บ)
vertices. Let ๐บ
ฬƒ The boundary of every
denote the number of regions in ๐บ.
region contains at least three edges and every edge is on the
boundary of at most two regions (the two regions are identical
when ๐‘’ is a cut-edge). For any edge ๐‘’ of ๐บ, let ๐‘Ÿ๐บ1 (๐‘’) and ๐‘Ÿ๐บ2 (๐‘’)
ฬƒ which
be the numbers of edges comprising the regions in ๐บ
the edge ๐‘’ borders. It is clear that every ๐‘’ = ๐‘ฅ๐‘ฆ โˆˆ ๐ธ(๐บ),
๓ต„จ
๓ต„จ
๐‘Ÿ๐บ2 (๐‘’) โฉพ ๐‘Ÿ๐บ1 (๐‘’) โฉพ 4 if ๓ต„จ๓ต„จ๓ต„จ๐‘๐บ (๐‘ฅ) โˆฉ ๐‘๐บ (๐‘ฆ)๓ต„จ๓ต„จ๓ต„จ = 0,
๓ต„จ
๓ต„จ
๐‘Ÿ๐บ2 (๐‘’) โฉพ 4, ๐‘Ÿ๐บ1 (๐‘’) โฉพ 3 if ๓ต„จ๓ต„จ๓ต„จ๐‘๐บ (๐‘ฅ) โˆฉ ๐‘๐บ (๐‘ฆ)๓ต„จ๓ต„จ๓ต„จ = 1,
(30)
๓ต„จ
๓ต„จ
๐‘Ÿ๐บ2 (๐‘’) โฉพ ๐‘Ÿ๐บ1 (๐‘’) โฉพ 3 if ๓ต„จ๓ต„จ๓ต„จ๐‘๐บ (๐‘ฅ) โˆฉ ๐‘๐บ (๐‘ฆ)๓ต„จ๓ต„จ๓ต„จ โฉพ 2.
Following Carlson and Develin [42], for any edge ๐‘’ = ๐‘ฅ๐‘ฆ of
๐บ, we define
๐ท๐บ (๐‘’) =
1
1
1
1
+
+
+
โˆ’ 1.
๐‘‘๐บ (๐‘ฅ) ๐‘‘๐บ (๐‘ฆ) ๐‘Ÿ๐บ1 (๐‘’) ๐‘Ÿ๐บ2 (๐‘’)
(31)
By the well-known Eulerโ€™s formula
๐œ (๐บ) โˆ’ ๐œ€ (๐บ) + ๐œ™ (๐บ) = 2 โˆ’ 2๐œŒ (๐บ) ,
(32)
it is easy to see that
โˆ‘ ๐ท๐บ (๐‘’) = ๐œ (๐บ) โˆ’ ๐œ€ (๐บ) + ๐œ™ (๐บ) = 2 โˆ’ 2๐œŒ (๐บ) .
๐‘’โˆˆ๐ธ(๐บ)
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If ๐บ is a planar graph, that is, ๐œŒ(๐บ) = 0, then
โˆ‘ ๐ท๐บ (๐‘’) = ๐œ (๐บ) โˆ’ ๐œ€ (๐บ) + ๐œ™ (๐บ) = 2.
๐‘’โˆˆ๐ธ(๐บ)
(34)
Combining these formulas with Theorem 18, Carlson and
Develin [42] gave a simple and intuitive proof of Theorem 74
and obtained the following result.
Theorem 96 (Carlson and Develin [42], 2006). Let ๐บ be a
connected graph which can be embedded on a torus. Then
๐‘(๐บ) โฉฝ ฮ”(๐บ) + 3.
Recently, Hou and Liu [50] improved Theorem 96 as
follows.
Theorem 97 (Hou and Liu [50], 2012). Let ๐บ be a connected
graph which can be embedded on a torus. Then ๐‘(๐บ) โฉฝ 9.
Moreover, if ฮ”(๐บ) =ฬธ 6, then ๐‘(๐บ) โฉฝ 8.
By the way, Cao et al. [51] generalized the result in
Theorem 79 to a connected graph ๐บ that can be embedded
on a torus; that is,
6,
{
{
{
{5,
๐‘ (๐บ) โ‰ค {
{
4,
{
{
{3,
if
if
if
if
๐‘” (๐บ) โฉพ 4 and ๐บ is not 4-regular,
๐‘” (๐บ) โฉพ 5,
๐‘” (๐บ) โฉพ 6 and ๐บ is not 3-regular,
๐‘” (๐บ) โฉพ 8.
(35)
Several authors used this method to obtain some results
on the bondage number. For example, Fischermann et al.
[46] used this method to prove the second conclusion in
Theorem 78. Recently, Cao et al. [52] have used this method to
deal with more general graphs with small crossing numbers.
First, they found the following property.
Lemma 98. ๐›ฟ(๐บ) โฉฝ 5 for any graph ๐บ with cr(๐บ) โฉฝ 5.
16
International Journal of Combinatorics
This result implies that ๐‘(๐บ) โฉฝ ฮ”(๐บ) + 4 by Corollary 15
for any graph ๐บ with cr(๐บ) โฉฝ 5. Cao et al. established the
following relation in terms of maximum planar subgraphs.
Lemma 99 (Cao et al. [52]). Let ๐บ be a connected graph with
crossing number cr(๐บ). For any maximum planar subgraph ๐ป
of ๐บ,
โˆ‘ ๐ท๐บ (๐‘’) โฉพ 2 โˆ’
๐‘’โˆˆ๐ธ(๐ป)
2cr (๐บ)
.
๐›ฟ (๐บ)
(36)
Using Carlson-Develin method and combining
Lemma 98 with Lemma 99, Cao et al. proved the following
results.
Theorem 100 (Cao et al. [52]). For any connected graph ๐บ,
๐‘(๐บ) โฉฝ ฮ”(๐บ) + 2 if ๐บ satisfies one of the following conditions:
(a) cr(๐บ) โฉฝ 3;
(b) cr(๐บ) = 4 and ๐บ is not 4-regular;
(c) cr(๐บ) = 5 and ๐บ contains no vertices of degree 4.
Theorem 101 (Cao et al. [52]). Let ๐บ be a connected graph with
ฮ”(๐บ) = 5 and cr(๐บ) โฉฝ 4. If no triangles contain two vertices of
degree 5, then ๐‘(๐บ) โฉฝ 6 = ฮ”(๐บ) + 1.
Theorem 102 (Cao et al. [52]). Let ๐บ be a connected graph
with ฮ”(๐บ) โฉพ 6 and cr(๐บ) โฉฝ 3. If ฮ”(๐บ) โฉพ 7 or if ฮ”(๐บ) = 6,
๐›ฟ(๐บ) =ฬธ 3 and every edge ๐‘’ = ๐‘ฅ๐‘ฆ with ๐‘‘๐บ(๐‘ฅ) = 5 and ๐‘‘๐บ(๐‘ฆ) = 6
is contained in at most one triangle, then ๐‘(๐บ) โฉฝ min{8, ฮ”(๐บ)+
1}.
Using Carlson-Develin method, it can be proved that
๐‘(๐บ) โฉฝ ฮ”(๐บ) + 2 if cr(๐บ) โฉฝ 3 (see the first conclusion in
Theorem 100), and not yet proved that ๐‘(๐บ) โฉฝ 8 if cr(๐บ) โฉฝ
3, although it has been proved by using other method (see
Corollary 90).
By using Carlson-Develin method, Samodivkin [25]
obtained some results on the bondage number for graphs with
some given properties.
Kim [53] showed that ๐‘(๐บ) โฉฝ ฮ”(๐บ) + 2 for a connected
graph ๐บ with genus 1, ฮ”(๐บ) โฉฝ 5 and having a toroidal
embedding of which at least one region is not 4-sided.
Recently, Gagarin and Zverovich [54] further have extended
Carlson-Develin ideas to establish a nice upper bound for
arbitrary graphs that can be embedded on orientable or
nonorientable topological surface.
Theorem 103 (Gagarin and Zverovich [54], 2012). Let ๐บ be
a graph embeddable on an orientable surface of genus โ„Ž and a
nonorientable surface of genus ๐‘˜. Then ๐‘(๐บ) โฉฝ min{ฮ”(๐บ) + โ„Ž +
2, ฮ”(๐บ) + ๐‘˜ + 1}.
This result generalizes the corresponding upper bounds
in Theorems 74 and 96 for any orientable or nonorientable topological surface. By investigating the proof of
Theorem 103, Huang [55] found that the issue of the orientability can be avoided by using the Euler characteristic
๐œ’(= ๐œ(๐บ) โˆ’ ๐œ€(๐บ) + ๐œ™(๐บ)) instead of the genera โ„Ž and ๐‘˜; the
relations are ๐œ’ = 2โˆ’2โ„Ž and ๐œ’ = 2โˆ’๐‘˜. To obtain the best result
from Theorem 103, one wants โ„Ž and ๐‘˜ as small as possible; this
is equivalent to having ๐œ’ as large as possible.
According to Theorem 103, if ๐บ is planar (โ„Ž = 0, ๐œ’ = 2) or
can be embedded on the real projective plane (๐‘˜ = 1, ๐œ’ = 1),
then ๐‘(๐บ) โฉฝ ฮ”(๐บ) + 2. In all other cases, Huang [55] had the
following improvement for Theorem 103; the proof is based
on the technique developed by Carlson-Develin and GagarinZverovich and includes some elementary calculus as a new
ingredient, mainly the intermediate-value theorem and the
mean-value theorem.
Theorem 104 (Huang [55], 2012). Let ๐บ be a graph embeddable on a surface whose Euler characteristic ๐œ’ is as large as
possible. If ๐œ’ โฉฝ 0, then ๐‘(๐บ) โฉฝ ฮ”(๐บ) + โŒŠ๐‘ŸโŒ‹, where ๐‘Ÿ is the largest
real root of the following cubic equation in ๐‘ง:
๐‘ง3 + 2๐‘ง2 + (6๐œ’ โˆ’ 7) ๐‘ง + 18๐œ’ โˆ’ 24 = 0.
(37)
In addition, if ๐œ’ decreases, then ๐‘Ÿ increases.
The following result is asymptotically equivalent to
Theorem 104.
Theorem 105 (Huang [55], 2012). Let ๐บ be a graph embeddable on a surface whose Euler characteristic ๐œ’ is as large as
possible. If ๐œ’ โฉฝ 0, then ๐‘(๐บ) < ฮ”(๐บ) + โˆš12 โˆ’ 6๐œ’ + 1/2, or
equivalently, ๐‘(๐บ) โฉฝ ฮ”(๐บ) + โŒˆโˆš12 โˆ’ 6๐œ’ โˆ’ 1/2โŒ‰.
Also, Gagarin and Zverovich [54] indicated that the upper
bound in Theorem 103 can be improved for larger values of
the genera โ„Ž and ๐‘˜ by adjusting the proofs and stated the
following general conjecture.
Conjecture 106 (Gagarin and Zverovich [54], 2012). For a
connected graph G of orientable genus โ„Ž and nonorientable
genus ๐‘˜,
๐‘ (๐บ) โฉฝ min {๐‘โ„Ž , ๐‘๐‘˜๓ธ€  , ฮ” (๐บ) + ๐‘œ (โ„Ž) , ฮ” (๐บ) + ๐‘œ (๐‘˜)} ,
(38)
where ๐‘โ„Ž and ๐‘๐‘˜๓ธ€  are constants depending, respectively, on the
orientable and nonorientable genera of ๐บ.
In the recent paper, Gagarin and Zverovich [56] provided
constant upper bounds for the bondage number of graphs on
topological surfaces, which can be used as the first estimation
for the constants ๐‘โ„Ž and ๐‘๐‘˜๓ธ€  of Conjecture 106.
Theorem 107 (Gagarin and Zverovich [56], 2013). Let ๐บ be
a connected graph of order ๐‘›. If ๐บ can be embedded on the
surface of its orientable or nonorientable genus of the Euler
characteristic ๐œ’, then
(a) ๐œ’ โฉพ 1 implies ๐‘(๐บ) โฉฝ 10;
(b) ๐œ’ โฉฝ 0 and ๐‘› > โˆ’12๐œ’ imply ๐‘(๐บ) โฉฝ 11;
(c) ๐œ’ โฉฝ โˆ’1 and ๐‘› โฉฝ โˆ’12๐œ’ imply ๐‘(๐บ) โฉฝ 11 +
3๐œ’(โˆš17 โˆ’ 8๐œ’ โˆ’ 3)/(๐œ’ โˆ’ 1) = 11 + ๐‘‚(โˆšโˆ’๐œ’).
Theorem 79 shows that a connected planar triangle-free
graph ๐บ has ๐‘(๐บ) โฉฝ 6. The following result generalizes to it
all the other topological surfaces.
International Journal of Combinatorics
17
Theorem 108 (Gagarin and Zverovich [56], 2013). Let ๐บ be a
connected triangle-free graph of order ๐‘›. If ๐บ can be embedded
on the surface of its orientable or nonorientable genus of the
Euler characteristic ๐œ’, then
(a) ๐œ’ โฉพ 1 implies ๐‘(๐บ) โฉฝ 6;
(b) ๐œ’ โฉฝ 0 and ๐‘› > โˆ’8๐œ’ imply ๐‘(๐บ) โฉฝ 7;
(c) ๐œ’ โฉฝ โˆ’1 and ๐‘› โฉฝ โˆ’8๐œ’ imply ๐‘(๐บ) โฉฝ 7โˆ’4๐œ’/(1+โˆš2 โˆ’ ๐œ’).
In [56], the authors also explicitly improved upper
bounds in Theorem 103 and gave tight lower bounds for
the number of vertices of graphs 2-cell embeddable on
topological surfaces of a given genus. The interested reader is
referred to the original paper for details. The following result
is interesting although its proof is easy.
Theorem 109 (Samodivkin [57]). Let ๐บ be a graph with order
๐‘› and girth ๐‘”. If ๐บ is embeddable on a surface whose Euler
characteristic ๐œ’ is as large as possible, then
๐‘ (๐บ) โฉฝ 3 +
4๐œ’๐‘”
8
.
โˆ’
๐‘” โˆ’ 2 ๐‘› (๐‘” โˆ’ 2)
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If ๐บ is a planar graph with girth ๐‘” โฉพ 4 + ๐‘– for any ๐‘– โˆˆ
{0, 1, 2}, then Theorem 109 leads to ๐‘(๐บ) โฉฝ 6 โˆ’ ๐‘–, which is
the result in Theorem 79 obtained by Fischermann et al. [46].
Also, in many cases the bound stated in Theorem 109 is better
than those given by Theorems 103 and 105.
8. Conditional Bondage Numbers
Since the concept of the bondage number is based upon the
domination, all sorts of dominations, which are variations
of the normal domination by adding restricted conditions
to the normal dominating set, can develop a new โ€œbondage
numberโ€ as long as a variation of domination number is
given. In this section, we survey results on the bondage
number under some restricted conditions.
8.1. Total Bondage Numbers. A dominating set ๐‘† of a graph
๐บ without isolated vertices is called total if the subgraph
induced by ๐‘† contains no isolated vertices. The minimum
cardinality of a total dominating set is called the total
domination number of ๐บ and denoted by ๐›พ๐‘‡(๐บ). It is clear
that ๐›พ(๐บ) โฉฝ ๐›พ๐‘‡(๐บ) โฉฝ 2๐›พ(๐บ) for any graph ๐บ without isolated
vertices.
The total domination in graphs was introduced by Cockayne et al. [58] in 1980. Pfaff et al. [59, 60] in 1983 showed
that the problem determining total domination number for
general graphs is NP-complete, even for bipartite graphs and
chordal graphs. Even now, total domination in graphs has
been extensively studied in the literature. In 2009, Henning
[61] gave a survey of selected recent results on total domination in graphs.
The total bondage number of ๐บ, denoted by ๐‘๐‘‡ (๐บ), is the
smallest cardinality of a subset ๐ต โŠ† ๐ธ(๐บ) with the property
that ๐บโˆ’๐ต contains no isolated vertices and ๐›พ๐‘‡ (๐บโˆ’๐ต) > ๐›พ๐‘‡ (๐บ).
From definition, ๐‘๐‘‡ (๐บ) may not exist for some graphs, for
example, ๐บ = ๐พ1,๐‘› . We put ๐‘๐‘‡ (๐บ) = โˆž if ๐‘๐‘‡ (๐บ) does not exist.
It is easy to see that ๐‘๐‘‡ (๐บ) is finite for any connected graph ๐บ
other than ๐พ1 , ๐พ2 , ๐พ3 , and ๐พ1,๐‘› . In fact, for such a graph ๐บ,
since there is a path of length 3 in ๐บ, we can find ๐ต1 โŠ† ๐ธ(๐บ)
such that ๐บ โˆ’ ๐ต1 is a spanning tree ๐‘‡ of ๐บ, containing a path
of length 3. So ๐›พ๐‘‡ (๐‘‡) = ๐›พ๐‘‡ (๐บ โˆ’ ๐ต1 ) โฉพ ๐›พ๐‘‡ (๐บ). For the tree ๐‘‡, we
find ๐ต2 โŠ† ๐ธ(๐‘‡) such that ๐›พ๐‘‡ (๐‘‡ โˆ’ ๐ต2 ) > ๐›พ๐‘‡ (๐‘‡) โฉพ ๐›พ๐‘‡(๐บ). Thus,
we have ๐›พ๐‘‡ (๐บโˆ’๐ต2 โˆ’๐ต1 ) > ๐›พ๐‘‡ (๐บ), and so ๐‘๐‘‡ (๐บ) โฉฝ |๐ต1 |+|๐ต2 |. In
the following discussion, we always assume that ๐‘๐‘‡ (๐บ) exists
when a graph ๐บ is mentioned.
In 1991, Kulli and Patwari [62] first studied the total
bondage number of a graph and calculated the exact values
of ๐‘๐‘‡ (๐บ) for some standard graphs.
Theorem 110 (Kulli and Patwari [62], 1991). For ๐‘› โฉพ 4,
3
๐‘๐‘‡ (๐ถ๐‘› ) = {
2
๐‘–๐‘“ ๐‘› โ‰ก 2 (mod 4) ,
๐‘œ๐‘กโ„Ž๐‘’๐‘Ÿ๐‘ค๐‘–๐‘ ๐‘’,
2
๐‘๐‘‡ (๐‘ƒ๐‘› ) = {
1
๐‘–๐‘“ ๐‘› โ‰ก 2 (mod 4) ,
๐‘œ๐‘กโ„Ž๐‘’๐‘Ÿ๐‘ค๐‘–๐‘ ๐‘’,
๐‘๐‘‡ (๐พ๐‘š,๐‘› ) = ๐‘š
(40)
๐‘ค๐‘–๐‘กโ„Ž 2 โฉฝ ๐‘š โฉฝ ๐‘›,
4
๐‘“๐‘œ๐‘Ÿ ๐‘› = 4,
๐‘๐‘‡ (๐พ๐‘› ) = {
2๐‘› โˆ’ 5 ๐‘“๐‘œ๐‘Ÿ ๐‘› โฉพ 5.
Recently, Hu et al. [63] have obtained some results on the
total bondage number of the Cartesian product ๐‘ƒ๐‘š ×๐‘ƒ๐‘› of two
paths ๐‘ƒ๐‘š and ๐‘ƒ๐‘› .
Theorem 111 (Hu et al. [63], 2012). For the Cartesian product
๐‘ƒ๐‘š × ๐‘ƒ๐‘› of two paths ๐‘ƒ๐‘š and ๐‘ƒ๐‘› ,
1 ๐‘–๐‘“ ๐‘› โ‰ก 0 (mod 3) ,
{
{
๐‘๐‘‡ (๐‘ƒ2 × ๐‘ƒ๐‘› ) = {2 ๐‘–๐‘“ ๐‘› โ‰ก 2 (mod 3) ,
{
{3 ๐‘–๐‘“ ๐‘› โ‰ก 1 (mod 3) ,
๐‘๐‘‡ (๐‘ƒ3 × ๐‘ƒ๐‘› ) = 1,
=1
{
{
{
{= 2
๐‘๐‘‡ (๐‘ƒ4 × ๐‘ƒ๐‘› ) {
{โฉฝ 3
{
{
{โฉฝ 4
๐‘–๐‘“
๐‘–๐‘“
๐‘–๐‘“
๐‘–๐‘“
(41)
๐‘› โ‰ก 1 (mod 5) ,
๐‘› โ‰ก 4 (mod 5) ,
๐‘› โ‰ก 2 (mod 5) ,
๐‘› โ‰ก 0, 3 (mod 5) .
Generalized Petersen graphs are an important class of
commonly used interconnection networks and have been
studied recently. By constructing a family of minimum total
dominating sets, Cao et al. [64] determined the total bondage
number of the generalized Petersen graphs.
From Theorem 6, we know that ๐‘(๐‘‡) โฉฝ 2 for a nontrivial
tree ๐‘‡. But given any positive integer ๐‘˜, Sridharan et al. [65]
constructed a tree ๐‘‡ for which ๐‘๐‘‡ (๐‘‡) = ๐‘˜. Let ๐ป๐‘˜ be the tree
obtained from a star ๐พ1,๐‘˜+1 by subdividing ๐‘˜ edges twice. The
tree ๐ป7 is shown in Figure 6. It can be easily verified that
๐‘๐‘‡ (๐ป๐‘˜ ) = ๐‘˜. This fact can be stated as the following theorem.
Theorem 112 (Sridharan et al. [65], 2007). For any positive
integer ๐‘˜, there exists a tree ๐‘‡ with ๐‘๐‘‡ (๐‘‡) = ๐‘˜.
18
International Journal of Combinatorics
gave an upper bound on ๐‘๐‘‡ (๐บ) in terms of the girth of
๐บ.
Theorem 117 (Rad and Raczek [66], 2011). ๐‘๐‘‡ (๐บ) โฉฝ 4ฮ”(๐บ) โˆ’ 5
for any graph ๐บ with ๐‘”(๐บ) โฉพ 4.
Figure 6: ๐ป7 .
Combining Theorem 1 with Theorem 112, we have what
follows.
Corollary 113 (Sridharan et al. [65], 2007). The bondage
number and the total bondage number are unrelated, even for
trees.
However, Sridharan et al. [65] gave an upper bound on
the total bondage number of a tree in terms of its maximum
degree. For any tree ๐‘‡ of order ๐‘›, if ๐‘‡ =ฬธ ๐พ1,๐‘›โˆ’1 , then ๐‘๐‘‡ (๐‘‡) =
min{ฮ”(๐‘‡), (1/3)(๐‘› โˆ’ 1)}. Rad and Raczek [66] improved this
upper bound and gave a constructive characterization of a
certain class of trees attaching the upper bound.
Theorem 114 (Rad and Raczek [66], 2011). ๐‘๐‘‡ (๐‘‡) โฉฝ ฮ”(๐‘‡) โˆ’ 1
for any tree ๐‘‡ with maximum degree at least three.
In general, the decision problem for ๐‘๐‘‡ (๐บ) is NP-complete
for any graph ๐บ. We state the decision problem for the total
bondage as follows.
Problem 3. Consider the decision problem:
Total Bondage Problem
Instance: a graph ๐บ and a positive integer ๐‘.
Question: is ๐‘๐‘‡ (๐บ) โฉฝ ๐‘?
Theorem 115 (Hu and Xu [16], 2012). The total bondage
problem is NP-complete.
Consequently, in view of its computational hardness,
it is significative to establish some sharp bounds on the
total bondage number of a graph in terms of other graphic
parameters. In 1991, Kulli and Patwari [62] showed that
๐‘๐‘‡ (๐บ) โฉฝ 2๐‘› โˆ’ 5 for any graph ๐บ with order ๐‘› โฉพ 5. Sridharan
et al. [65] improved this result as follows.
Theorem 116 (Sridharan et al. [65], 2007). For any graph ๐บ
with order ๐‘›, if ๐‘› โฉพ 5, then
1
{
๐‘–๐‘“ ๐บ ๐‘๐‘œ๐‘›๐‘ก๐‘Ž๐‘–๐‘›๐‘  ๐‘›๐‘œ ๐‘๐‘ฆ๐‘๐‘™๐‘’๐‘ ,
(๐‘› โˆ’ 1)
{
{3
(42)
๐‘๐‘‡ (๐บ) โฉฝ {๐‘› โˆ’ 1
๐‘–๐‘“ ๐‘” (๐บ) โฉพ 5,
{
{
๐‘–๐‘“ ๐‘” (๐บ) = 4.
{๐‘› โˆ’ 2
Rad and Raczek [66] also established some upper
bounds on ๐‘๐‘‡ (๐บ) for a general graph ๐บ. In particular, they
8.2. Paired Bondage Numbers. A dominating set ๐‘† of ๐บ is
called to be paired if the subgraph induced by ๐‘† contains
a perfect matching. The paired domination number of ๐บ,
denoted by ๐›พ๐‘ƒ (๐บ), is the minimum cardinality of a paired
dominating set of ๐บ. Clearly, ๐›พ๐‘‡ (๐บ) โฉฝ ๐›พ๐‘ƒ (๐บ) for every
connected graph ๐บ with order at least two, where ๐›พ๐‘‡(๐บ) is
the total domination number of ๐บ, and ๐›พ๐‘ƒ (๐บ) โฉฝ 2๐›พ(๐บ) for
any graph ๐บ without isolated vertices. Paired domination was
introduced by Haynes and Slater [67, 68] and further studied
in [69โ€“72].
The paired bondage number of ๐บ with ๐›ฟ(๐บ) โฉพ 1, denoted
by ๐‘P (๐บ), is the minimum cardinality among all sets of edges
๐ต โŠ† ๐ธ such that ๐›ฟ(๐บ โˆ’ ๐ต) โฉพ 1 and ๐›พ๐‘ƒ (๐บ โˆ’ ๐ต) > ๐›พ๐‘ƒ (๐บ).
The concept of the paired bondage number was first
proposed by Raczek [73] in 2008. The following observations
follow immediately from the definition of the paired bondage
number.
Observation 2. Let ๐บ be a graph with ๐›ฟ(๐บ) โฉพ 1.
(a) If ๐ป is a subgraph of ๐บ such that ๐›ฟ(๐ป) โฉพ 1, then
๐‘๐‘ƒ (๐ป) โฉฝ ๐‘๐‘ƒ (๐บ).
(b) If ๐ป is a subgraph of ๐บ such that ๐‘๐‘ƒ (๐ป) = 1 and ๐‘˜
is the number of edges removed to form ๐ป, then 1 โฉฝ
๐‘๐‘ƒ (๐บ) โฉฝ ๐‘˜ + 1.
(c) If ๐‘ฅ๐‘ฆ โˆˆ ๐ธ(๐บ) such that ๐‘‘๐บ(๐‘ฅ), ๐‘‘๐บ(๐‘ฆ) > 1, and ๐‘ฅ๐‘ฆ
belongs to each perfect matching of each minimum
paired dominating set of ๐บ, then ๐‘๐‘ƒ (๐บ) = 1.
Based on these observations, ๐‘๐‘ƒ (๐‘ƒ๐‘› ) โฉฝ 2. In fact, the
paired bondage number of a path has been determined.
Theorem 118 (Raczek [73], 2008). For any positive integer ๐‘˜,
if ๐‘› โฉพ 2, then
0 ๐‘–๐‘“ ๐‘› = 2, 3, ๐‘œ๐‘Ÿ 5,
{
{
๐‘๐‘ƒ (๐‘ƒ๐‘› ) = {1 ๐‘–๐‘“ ๐‘› = 4๐‘˜, 4๐‘˜ + 3, ๐‘œ๐‘Ÿ 4๐‘˜ + 6,
{
{2 ๐‘œ๐‘กโ„Ž๐‘’๐‘Ÿ๐‘ค๐‘–๐‘ ๐‘’.
(43)
For a cycle ๐ถ๐‘› of order ๐‘› โฉพ 3, since ๐›พ๐‘ƒ (๐ถ๐‘› ) = ๐›พ๐‘ƒ (๐‘ƒ๐‘› ), from
Theorem 118 the following result holds immediately.
Corollary 119 (Raczek [73], 2008). For any positive integer ๐‘˜,
if ๐‘› โฉพ 3, then
0 ๐‘–๐‘“ ๐‘› = 3 ๐‘œ๐‘Ÿ 5,
{
{
๐‘๐‘ƒ (๐ถ๐‘› ) = {2 ๐‘–๐‘“ ๐‘› = 4๐‘˜, 4๐‘˜ + 3, ๐‘œ๐‘Ÿ 4๐‘˜ + 6,
{
{3 ๐‘œ๐‘กโ„Ž๐‘’๐‘Ÿ๐‘ค๐‘–๐‘ ๐‘’.
(44)
A wheel ๐‘Š๐‘› , where ๐‘› โฉพ 4, is a graph with ๐‘› vertices,
formed by connecting a single vertex to all vertices of a cycle
๐ถ๐‘›โˆ’1 . Of course ๐›พ๐‘ƒ (๐‘Š๐‘› ) = 2.
International Journal of Combinatorics
19
๐ด
.
..
.
..
๐ต
๐ด
๐‘˜
๐ด
๐ด
๐ต
๐ต
๐ถ
๐ต
๐ถ
๐ด
๐ด
Figure 7: A tree ๐‘‡ with ๐‘๐‘ƒ (๐‘‡) = ๐‘˜.
Figure 8: A tree ๐‘‡ belong to the family T.
Theorem 120 (Raczek [73], 2008). For positive integers ๐‘› and
๐‘š,
Theorem 123 (Raczek [73], 2008). For a tree ๐‘‡, ๐‘๐‘ƒ (๐‘‡) = 0 if
and only if ๐‘‡ is in T.
๐‘๐‘ƒ (๐พ๐‘š,๐‘› ) = ๐‘š ๐‘“๐‘œ๐‘Ÿ 1 < ๐‘š โฉฝ ๐‘›,
4 ๐‘–๐‘“ ๐‘› = 4,
{
{
๐‘๐‘ƒ (๐‘Š๐‘› ) = {3 ๐‘–๐‘“ ๐‘› = 5,
{
{2 ๐‘œ๐‘กโ„Ž๐‘’๐‘Ÿ๐‘ค๐‘–๐‘ ๐‘’.
We state the decision problem for the paired bondage as
follows.
(45)
Theorem 16 shows that if ๐‘‡ is a nontrivial tree, then
๐‘(๐‘‡) โฉฝ 2. However, no similar result exists for paired
bondage. For any nonnegative integer ๐‘˜, let ๐‘‡๐‘˜ be a tree
obtained by subdividing all but one edge of a star ๐พ1,๐‘˜+1 (as
shown in Figure 7). It is easy to see that ๐‘๐‘ƒ (๐‘‡๐‘˜ ) = ๐‘˜. We
state this fact as the following theorem, which is similar to
Theorem 112.
Theorem 121 (Raczek [73], 2008). For any nonnegative integer
๐‘˜, there exists a tree ๐‘‡ with ๐‘๐‘ƒ (๐‘‡) = ๐‘˜.
Consider the tree defined in Figure 5 for ๐‘˜ = 3. Then
๐‘(๐‘‡3 ) โฉฝ 2 and ๐‘๐‘ƒ (๐‘‡3 ) = 3. On the other hand, ๐‘(๐‘ƒ4 ) = 2
and ๐‘๐‘ƒ (๐‘ƒ4 ) = 1. Thus, we have what follows, which is similar
to Corollary 113.
Corollary 122. The bondage number and the paired bondage
number are unrelated, even for trees.
A constructive characterization of trees with ๐‘(๐‘‡) = 2
is given by Hartnell and Rall in [12]. Raczek [73] provided
a constructive characterization of trees with ๐‘๐‘ƒ (๐‘‡) = 0. In
order to state the characterization, we define a labeling and
three simple operations on a tree ๐‘‡. Let ๐‘ฆ โˆˆ ๐‘‰(๐‘‡) and let โ„“(๐‘ฆ)
be the label assigned to ๐‘ฆ.
Operation T1 . If โ„“(๐‘ฆ) = ๐ต, add a vertex ๐‘ฅ and the
edge ๐‘ฆ๐‘ฅ, and let โ„“(๐‘ฅ) = ๐ด.
Operation T2 . If โ„“(๐‘ฆ) = ๐ถ, add a path (๐‘ฅ1 , ๐‘ฅ2 ) and the
edge ๐‘ฆ๐‘ฅ1 , and let โ„“(๐‘ฅ1 ) = ๐ต, โ„“(๐‘ฅ2 ) = ๐ด.
Operation T3 . If โ„“(๐‘ฆ) = ๐ต, add a path (๐‘ฅ1 , ๐‘ฅ2 , ๐‘ฅ3 )
and the edge ๐‘ฆ๐‘ฅ1 , and let โ„“(๐‘ฅ1 ) = ๐ถ, โ„“(๐‘ฅ2 ) = ๐ต, and
โ„“(๐‘ฅ3 ) = ๐ด.
Let ๐‘ƒ2 = (๐‘ข, V) with โ„“(๐‘ข) = ๐ด and โ„“(V) = ๐ต. Let T be
the class of all trees obtained from the labeled ๐‘ƒ2 by a finite
sequence of operations T1 , T2 , T3 .
A tree ๐‘‡ in Figure 8 belongs to the family T.
Raczek [73] obtained the following characterization of all
trees ๐‘‡ with ๐‘๐‘ƒ (๐‘‡) = 0.
Problem 4. Consider the decision problem:
Paired Bondage Problem
Instance: a graph ๐บ and a positive integer ๐‘.
Question: is ๐‘๐‘ƒ (๐บ) โฉฝ ๐‘?
Conjecture 124. The paired bondage problem is NP-complete.
8.3. Restrained Bondage Numbers. A dominating set ๐‘† of ๐บ is
called to be restrained if the subgraph induced by ๐‘† contains
no isolated vertices, where ๐‘† = ๐‘‰(๐บ) \ ๐‘†. The restrained
domination number of ๐บ, denoted by ๐›พ๐‘… (๐บ), is the smallest
cardinality of a restrained dominating set of ๐บ. It is clear that
๐›พ๐‘… (๐บ) exists and ๐›พ(๐บ) โฉฝ ๐›พ๐‘… (๐บ) for any nonempty graph ๐บ.
The concept of restrained domination was introduced
by Telle and Proskurowski [74] in 1997, albeit indirectly, as
a vertex partitioning problem. Concerning the study of the
restrained domination numbers, the reader is referred to [74โ€“
79].
In 2008, Hattingh and Plummer [80] defined the
restrained bondage number ๐‘R (๐บ) of a nonempty graph ๐บ to
be the minimum cardinality among all subsets ๐ต โŠ† ๐ธ(๐บ) for
which ๐›พ๐‘… (๐บ โˆ’ ๐ต) > ๐›พ๐‘… (๐บ).
For some simple graphs, their restrained domination
numbers can be easily determined, and so restrained bondage
numbers have been also determined. For example, it is clear
that ๐›พ๐‘… (๐พ๐‘› ) = 1. Domke et al. [77] showed that ๐›พ๐‘… (๐ถ๐‘› ) =
๐‘› โˆ’ 2โŒŠ๐‘›/3โŒ‹ for ๐‘› โฉพ 3, and ๐›พ๐‘… (๐‘ƒ๐‘› ) = ๐‘› โˆ’ 2โŒŠ(๐‘› โˆ’ 1)/3โŒ‹ for ๐‘› โฉพ 1.
Using these results, Hattingh and Plummer [80] obtained the
restricted bondage numbers for ๐พ๐‘› , ๐ถ๐‘› , ๐‘ƒ๐‘› , and ๐บ = ๐พ๐‘›1 ,๐‘›2 ,...,๐‘›๐‘ก
as follows.
Theorem 125 (Hattingh and Plummer [80], 2008). For ๐‘› โฉพ 3,
{1
๐‘R (๐พ๐‘› ) = { ๐‘›
โŒˆ โŒ‰
{ 2
๐‘R (๐ถ๐‘› ) = {
๐‘–๐‘“ ๐‘› = 3,
๐‘œ๐‘กโ„Ž๐‘’๐‘Ÿ๐‘ค๐‘–๐‘ ๐‘’,
1 ๐‘–๐‘“ ๐‘› โ‰ก 0 (mod 3) ,
2 ๐‘œ๐‘กโ„Ž๐‘’๐‘Ÿ๐‘ค๐‘–๐‘ ๐‘’,
๐‘R (๐‘ƒ๐‘› ) = 1,
๐‘› โฉพ 4,
20
International Journal of Combinatorics
๐‘R (๐บ)
We now consider the relation between ๐‘๐‘… (๐บ) and ๐‘(๐บ).
๐‘š
{
โŒˆ โŒ‰
{
{
2
{
{
{
{2๐‘ก โˆ’ 2
{
= {2
{
{
{
๐‘กโˆ’1
{
{
{โˆ‘๐‘› โˆ’ 1
{
๐‘–
{๐‘–=1
๐‘–๐‘“ ๐‘›๐‘š = 1 ๐‘Ž๐‘›๐‘‘ ๐‘›๐‘š+1 โฉพ 2 (1 โฉฝ ๐‘š < ๐‘ก) ,
๐‘–๐‘“ ๐‘›1 = ๐‘›2 = โ‹… โ‹… โ‹… = ๐‘›๐‘ก = 2 (๐‘ก โฉพ 2) ,
๐‘–๐‘“ ๐‘›1 = 2 ๐‘Ž๐‘›๐‘‘ ๐‘›2 โฉพ 3 (๐‘ก = 2) ,
๐‘œ๐‘กโ„Ž๐‘’๐‘Ÿ๐‘ค๐‘–๐‘ ๐‘’.
(46)
Theorem 126 (Hattingh and Plummer [80], 2008). For a tree
๐‘‡ of order ๐‘› (โฉพ 4), ๐‘‡ โ‰‡ ๐พ1,๐‘›โˆ’1 if and only if ๐‘R (๐‘‡) = 1.
Theorem 126 shows that the restrained bondage number
of a tree can be computed in constant time. However, the
decision problem for ๐‘R (๐บ) is NP-complete even for bipartite
graphs.
Problem 5. Consider the decision problem:
Restrained Bondage Problem
Instance: a graph ๐บ and a positive integer ๐‘.
Question: is ๐‘R (๐บ) โฉฝ ๐‘?
Theorem 127 (Hattingh and Plummer [80], 2008). The
restrained bondage problem is NP-complete, even for bipartite
graphs.
Consequently, in view of its computational hardness,
it is significative to establish some sharp bounds on the
restrained bondage number of a graph in terms of other
graphic parameters.
Theorem 128 (Hattingh and Plummer [80], 2008). For any
graph ๐บ with ๐›ฟ(๐บ) โฉพ 2,
๐‘R (๐บ) โฉฝ min {๐‘‘๐บ (๐‘ฅ) + ๐‘‘๐บ (๐‘ฆ) โˆ’ 2} .
๐‘ฅ๐‘ฆโˆˆ๐ธ(๐บ)
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Corollary 129. ๐‘R (๐บ) โฉฝ ฮ”(๐บ) + ๐›ฟ(๐บ) โˆ’ 2 for any graph ๐บ with
๐›ฟ(๐บ) โฉพ 2.
Notice that the bounds stated in Theorem 128 and
Corollary 129 are sharp. Indeed the class of cycles whose
orders are congruent to 1, 2 (mod 3) have a restrained bondage
number achieving these bounds (see Theorem 125).
Theorem 128 is an analogue of Theorem 14. A quite
natural problem is whether or not, for restricted bondage
number, there are analogues of Theorem 16, Theorem 18,
Theorem 29, and so on. Indeed, we should consider all results
for the bondage number whether or not there are analogues
for restricted bondage number.
Theorem 130 (Hattingh and Plummer [80], 2008). For any
graph ๐บ with ๐›พ๐‘… (๐บ) = 2,
๐‘R (๐บ) โฉฝ ฮ” (๐บ) + 1.
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Theorem 131 (Hattingh and Plummer [80], 2008). If ๐›พ๐‘… (๐บ) =
๐›พ(๐บ) for some graph ๐บ, then ๐‘R (๐บ) โฉฝ ๐‘(๐บ).
Corollary 129 and Theorem 131 provide a possibility to
attack Conjecture 73 by characterizing planar graphs with
๐›พ๐‘… = ๐›พ and ๐‘R = ๐‘ when 3 โฉฝ ฮ” โฉฝ 6.
However, we do not have ๐‘R (๐บ) = ๐‘(๐บ) for any graph
๐บ even if ๐›พ๐‘… (๐บ) = ๐›พ(๐บ). Observe that ๐›พ๐‘… (๐พ3 ) = ๐›พ(๐พ3 ), yet
๐‘๐‘… (๐พ3 ) = 1 and ๐‘(๐พ3 ) = 2. We still may not claim that
๐‘๐‘… (๐บ) = ๐‘(๐บ) even in the case that every ๐›พ-set is a ๐›พ๐‘… -set.
The example ๐พ3 again demonstrates this.
Theorem 132 (Hattingh and Plummer [80], 2008). There is
an infinite class of graphs in which each graph ๐บ satisfies ๐‘(๐บ) <
๐‘R (๐บ).
In fact, such an infinite class of graphs can be obtained as
follows. Let ๐ป be a connected graph, BC(๐ป) a graph obtained
from ๐ป by attaching โ„“ (โฉพ 2) vertices of degree 1 to each
vertex in ๐ป, and B = {๐บ : ๐บ = BC(๐ป) for some graph ๐ป
such that ๐›ฟ(๐ป) โฉพ 2}. By Theorem 3, we can easily check that
๐‘(๐บ) = 1 < 2 โฉฝ min{๐›ฟ(๐ป), โ„“} = ๐‘R (๐บ) for any ๐บ โˆˆ B.
Moreover, there exists a graph ๐บ โˆˆ B such that ๐‘R (๐บ) can
be much larger than ๐‘(๐บ), which is stated as the following
theorem.
Theorem 133 (Hattingh and Plummer [80], 2008). For each
positive integer ๐‘˜, there is a graph ๐บ such that ๐‘˜ = ๐‘R (๐บ)โˆ’๐‘(๐บ).
Combining Theorem 133 with Theorem 132, we have the
following corollary.
Corollary 134. The bondage number and the restrained
bondage number are unrelated.
Very recently, Jafari Rad et al. [81] have considered
the total restrained bondage in graphs based on both total
dominating sets and restrained dominating sets and obtained
several properties, exact values, and bounds for the total
restrained bondage number of a graph.
A restrained dominating set ๐‘† of a graph ๐บ without isolated vertices is called to be a total restrained dominating set
if ๐‘† is a total dominating set. The total restrained domination
number of ๐บ, denoted by ๐›พ TR (๐บ), is the minimum cardinality
of a total restrained dominating set of ๐บ. For references
on this domination in graphs, see [3, 61, 82โ€“85]. The total
restrained bondage number ๐‘ TR (๐บ) of a graph ๐บ with no
isolated vertex, is the cardinality of a smallest set of edges ๐ต โŠ†
๐ธ(๐บ) for which ๐บโˆ’๐ต has no isolated vertex and ๐›พ TR (๐บโˆ’๐ต) >
๐›พ TR (๐บ). In the case that there is no such subset ๐ต, we define
๐‘ TR (๐บ) = โˆž.
Ma et al. [84] determined that ๐›พ TR (๐‘ƒ๐‘› ) = ๐‘› โˆ’ 2โŒŠ(๐‘› โˆ’ 2)/4โŒ‹
for ๐‘› โฉพ 3, ๐›พ TR (๐ถ๐‘› ) = ๐‘› โˆ’ 2โŒŠ๐‘›/4โŒ‹ for ๐‘› โฉพ 3, ๐›พ TR (๐พ3 ) =
3 and ๐›พ TR (๐พ๐‘› ) = 2 for ๐‘› =ฬธ 3, and ๐›พ TR (๐พ1,๐‘› ) = 1 + ๐‘›
and ๐›พ TR (๐พ๐‘š,๐‘› ) = 2 for 2 โฉฝ ๐‘š โฉฝ ๐‘›. According to these
results, Jafari Rad et al. [81] determined the exact values
of total restrained bondage numbers for the corresponding
graphs.
International Journal of Combinatorics
21
Theorem 135 (Jafari Rad et al. [81]). For an integer ๐‘›,
โˆž
๐‘ TR (๐‘ƒ๐‘› ) = {
1,
๐‘ TR
โˆž
{
{
(๐ถ๐‘› ) = {1,
{
{2,
๐‘–๐‘“ 2 โฉฝ ๐‘› โฉฝ 5,
๐‘–๐‘“ ๐‘› โฉพ 5
๐‘–๐‘“ ๐‘› = 3,
๐‘–๐‘“ 3 < ๐‘›, ๐‘› โ‰ก 0, 1 (mod 4)
๐‘–๐‘“ 3 < ๐‘›, ๐‘› โ‰ก 2, 3 (mod 4)
๐‘ TR (๐‘Š๐‘› ) = 2
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๐‘“๐‘œ๐‘Ÿ ๐‘› โฉพ 6,
๐‘ TR (๐พ๐‘› ) = ๐‘› โˆ’ 1 ๐‘“๐‘œ๐‘Ÿ ๐‘› โฉพ 4,
๐‘ TR (๐พ๐‘š,๐‘› ) = ๐‘š โˆ’ 1
๐‘“๐‘œ๐‘Ÿ 2 โฉฝ ๐‘š โฉฝ ๐‘›.
๐‘ TR (๐พ๐‘› ) = ๐‘› โˆ’ 1 shows the fact that for any integer ๐‘› โฉพ 3 there
exists a connected graph, namely, ๐พ๐‘›+1 such that ๐‘ TR (๐พ๐‘›+1 ) =
๐‘›; that is, the total restrained bondage number is unbounded
from the above.
A vertex is said to be a support vertex if it is adjacent to
a vertex of degree one. Let A be the family of all connected
graphs such that ๐บ belongs to A if and only if every edge of ๐บ
is incident to a support vertex or ๐บ is a cycle of length three.
Proposition 136 (Cyman and Raczek [82], 2006). For a
connected graph ๐บ of order ๐‘›, ๐›พ TR (๐บ) = ๐‘› if and only if ๐บ โˆˆ A.
Hence, if ๐บ โˆˆ A, then the removal of any subset of edges
of ๐บ cannot increase the total restrained domination number
of ๐บ and thus ๐‘ TR (๐บ) = โˆž. When ๐บ โˆ‰ A, a result similar to
Theorem 6 is obtained.
Theorem 137 (Jafari Rad et al. [81]). For any tree ๐‘‡ โˆ‰ A,
๐‘ TR (๐‘‡) โฉฝ 2. This bound is sharp.
In the same paper, the authors provided a characterization
for trees ๐‘‡ โˆ‰ A with ๐‘ TR (๐‘‡) = 1 or 2. The following result is
similar to Observation 1.
Observation 3 (Jafari Rad et al. [81]). If ๐‘˜ edges can be
removed from a graph ๐บ to obtain a graph ๐ป without an
isolated vertex and with ๐‘ TR (๐ป) = ๐‘ก, then ๐‘ TR (๐บ) โฉฝ ๐‘˜ + ๐‘ก.
Applying Theorem 137 and Observation 3, Jafari Rad et al.
characterized all connected graphs ๐บ with ๐‘ TR (๐บ) = โˆž.
Theorem 138 (Jafari Rad et al. [81]). For a connected graph ๐บ,
๐‘ TR (๐บ) = โˆž if and only if ๐บ โˆˆ A.
8.4. Other Conditional Bondage Numbers. A subset ๐ผ โŠ† ๐‘‰(๐บ)
is called an independent set if no two vertices in ๐ผ are adjacent
in ๐บ. The maximum cardinality among all independent sets
is called the independence number of ๐บ, denoted by ๐›ผ(๐บ).
A dominating set ๐‘† of a graph ๐บ is called to be independent if ๐‘† is an independent set of ๐บ. The minimum
cardinality among all independent dominating set is called
the independence domination number of ๐บ and denoted by
๐›พ๐ผ(๐บ).
Since an independent dominating set is not only a
dominating set but also an independent set, ๐›พ(๐บ) โฉฝ ๐›พ๐ผ (๐บ) โฉฝ
๐›ผ(๐บ) for any graph ๐บ.
It is clear that a maximal independent set is certainly a
dominating set. Thus, an independent set is maximal if and
only if it is an independent dominating set, and so ๐›พ๐ผ (๐บ) is the
minimum cardinality among all maximal independent sets
of ๐บ. This graph-theoretical invariant has been well studied
in the literature; see, for example, Haynes et al. [3] for early
results, Goddard and Henning [86] for recent results on
independent domination in graphs.
In 2003, Zhang et al. [87] defined the independence
bondage number ๐‘๐ผ (๐บ) of a nonempty graph ๐บ to be the
minimum cardinality among all subsets ๐ต โŠ† ๐ธ(๐บ) for which
๐›พ๐ผ (๐บ โˆ’ ๐ต) > ๐›พ๐ผ (๐บ). For some ordinary graphs, their independence domination numbers can be easily determined, and so
independence bondage numbers have been also determined.
Clearly, ๐‘๐ผ (๐พ๐‘› ) = 1 if ๐‘› โฉพ 2.
Theorem 139 (Zhang et al. [87], 2003). For any integer ๐‘› โฉพ 4,
๐‘๐ผ (๐ถ๐‘› ) = {
1 ๐‘–๐‘“ ๐‘› โ‰ก 1 (mod 2) ,
2 ๐‘–๐‘“ ๐‘› โ‰ก 0 (mod 2) ,
1 ๐‘–๐‘“ ๐‘› โ‰ก 0 (mod 2) ,
๐‘๐ผ (๐‘ƒ๐‘› ) = {
2 ๐‘–๐‘“ ๐‘› โ‰ก 1 (mod 2) ,
๐‘๐ผ (๐พ๐‘š,๐‘› ) = ๐‘›
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๐‘“๐‘œ๐‘Ÿ ๐‘š โฉฝ ๐‘›.
Apart from the above-mentioned results, as far as we find
no other results on the independence bondage number in the
present literature, we never knew much of any result on this
parameter for a tree.
A subset ๐‘† of ๐‘‰(๐บ) is called an equitable dominating
set if for every ๐‘ฆ โˆˆ ๐‘‰(๐บ โˆ’ ๐‘†) there exists a vertex ๐‘ฅ โˆˆ ๐‘†
such that ๐‘ฅ๐‘ฆ โˆˆ ๐ธ(๐บ) and |๐‘‘๐บ(๐‘ฅ) โˆ’ ๐‘‘๐บ(๐‘ฆ)| โฉฝ 1 proposed
by Dharmalingam [88]. The minimum cardinality of such a
dominating set is called the equitable domination number and
is denoted by ๐›พ๐ธ (๐บ). Deepak et al. [89] defined the equitable
bondage number ๐‘๐ธ (๐บ) of a graph ๐บ to be the cardinality of a
smallest set ๐ต โŠ† ๐ธ(๐บ) of edges for which ๐›พ๐ธ (๐บ โˆ’ ๐ต) > ๐›พ๐ธ (๐บ)
and determined ๐‘๐ธ (๐บ) for some special graphs.
Theorem 140 (Deepak et al. [89], 2011). For any integer ๐‘› โฉพ 2,
๐‘›
๐‘๐ธ (๐พ๐‘› ) = โŒˆ โŒ‰ ,
2
๐‘๐ธ (๐พ๐‘š,๐‘› ) = ๐‘š
๐‘“๐‘œ๐‘Ÿ 0 โฉฝ ๐‘› โˆ’ ๐‘š โฉฝ 1,
3 ๐‘–๐‘“ ๐‘› โ‰ก 1 (mod 3)
๐‘๐ธ (๐ถ๐‘› ) = {
2 ๐‘œ๐‘กโ„Ž๐‘’๐‘Ÿ๐‘ค๐‘–๐‘ ๐‘’,
2
๐‘๐ธ (๐‘ƒ๐‘› ) = {
1
๐‘–๐‘“ ๐‘› โ‰ก 1 (mod 3)
๐‘œ๐‘กโ„Ž๐‘’๐‘Ÿ๐‘ค๐‘–๐‘ ๐‘’,
๐‘๐ธ (๐‘‡) โฉฝ 2 ๐‘“๐‘œ๐‘Ÿ ๐‘Ž๐‘›๐‘ฆ ๐‘›๐‘œ๐‘›๐‘ก๐‘Ÿ๐‘–๐‘ฃ๐‘–๐‘Ž๐‘™ ๐‘ก๐‘Ÿ๐‘’๐‘’ ๐‘‡,
๐‘๐ธ (๐บ) โฉฝ ๐‘› โˆ’ 1 ๐‘“๐‘œ๐‘Ÿ ๐‘Ž๐‘›๐‘ฆ ๐‘๐‘œ๐‘›๐‘›๐‘’๐‘๐‘ก๐‘’๐‘‘ ๐‘”๐‘Ÿ๐‘Ž๐‘โ„Ž ๐บ ๐‘œ๐‘“ ๐‘œ๐‘Ÿ๐‘‘๐‘’๐‘Ÿ ๐‘›.
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22
International Journal of Combinatorics
As far as we know, there are no more results on the
equitable bondage number of graphs in the present literature.
There are other variations of the normal domination by
adding restricted conditions to the normal dominating set.
For example, the connected domination set of a graph ๐บ, proposed by Sampathkumar and Walikar [90], is a dominating
set ๐‘† such that the subgraph induced by ๐‘† is connected. The
problem of finding a minimum connected domination set in
a graph is equivalent to the problem of finding a spanning tree
with maximum number of vertices of degree one [91], and has
some important applications in wireless networks [92]; thus,
it has received much research attention. However, for such
an important domination, there is no result on its bondage
number. We believe that the topic is of significance for future
research.
Theorem 142 (Lu and Xu [96], 2011). Let ๐‘‡ be a tree with
ฮ”(๐‘‡) โฉพ ๐‘ โฉพ 2. Then ๐‘๐‘ (๐‘‡) = 1 if and only if for any ๐›พ๐‘ -set
๐‘† of ๐‘‡ there exists an edge ๐‘ฅ๐‘ฆ โˆˆ (๐‘†, ๐‘†) such that ๐‘ฆ โˆˆ ๐‘๐‘ (๐‘ฅ, ๐‘†, ๐‘‡)
The notation ๐‘†(๐‘Ž, ๐‘) denotes a double star obtained by
adding an edge between the central vertices of two stars ๐พ1,๐‘Žโˆ’1
and ๐พ1,๐‘โˆ’1 . And the vertex with degree ๐‘Ž (resp., ๐‘) in ๐‘†(๐‘Ž, ๐‘) is
called the ๐ฟ-central vertex (resp., ๐‘…-central vertex) of ๐‘†(๐‘Ž, ๐‘):
To characterize all trees attaining the upper bound given
in Theorem 141, we define three types of operations on a tree
๐‘‡ with ฮ”(๐‘‡) = ฮ” โฉพ ๐‘ + 1:
Type 1: attach a pendant edge to a vertex ๐‘ฆ with ๐‘‘๐‘‡ (๐‘ฆ) โฉฝ ๐‘โˆ’
2 in ๐‘‡.
9. Generalized Bondage Numbers
There are various generalizations of the classical domination,
such as ๐‘-domination, distance domination, fractional domination, Roman domination, and rainbow domination. Every
such a generalization can lead to a corresponding bondage.
In this section, we introduce some of them.
9.1. ๐‘-Bondage Numbers. In 1985, Fink and Jacobson [93]
introduced the concept of ๐‘-domination. Let ๐‘ be a positive
integer. A subset ๐‘† of ๐‘‰(๐บ) is a ๐‘-dominating set of ๐บ if
|๐‘† โˆฉ ๐‘๐บ(๐‘ฆ)| โฉพ ๐‘ for every ๐‘ฆ โˆˆ ๐‘†. The ๐‘-domination number
๐›พ๐‘ (๐บ) is the minimum cardinality among all ๐‘-dominating
sets of ๐บ. Any ๐‘-dominating set of ๐บ with cardinality ๐›พ๐‘ (๐บ)
is called a ๐›พ๐‘ -set of ๐บ. Note that a ๐›พ1 -set is a classic minimum
dominating set. Notice that every graph has a ๐‘-dominating
set since the vertex-set ๐‘‰(๐บ) is such a set. We also note that a 1dominating set is a dominating set, and so ๐›พ(๐บ) = ๐›พ1 (๐บ). The
๐‘-domination number has received much research attention;
see a state-of-the-art survey article by Chellali et al. [94].
It is clear from definition that every ๐‘-dominating set
of a graph certainly contains all vertices of degree at most
๐‘ โˆ’ 1. By this simple observation, to avoid the trivial case
occurrence, we always assume that ฮ”(๐บ) โฉพ ๐‘. For ๐‘ โฉพ 2, Lu
et al. [95] gave a constructive characterization of trees with
unique minimum ๐‘-dominating sets.
Recently, Lu and Xu [96] have introduced the concept
to the ๐‘-bondage number of ๐บ, denoted by ๐‘๐‘ (๐บ), as the
minimum cardinality among all sets of edges ๐ต โŠ† ๐ธ(๐บ) such
that ๐›พ๐‘ (๐บ โˆ’ ๐ต) > ๐›พ๐‘ (๐บ). Clearly, ๐‘1 (๐บ) = ๐‘(๐บ).
Lu and Xu [96] established a lower bound and an upper
bound on ๐‘๐‘ (๐‘‡) for any integer ๐‘ โฉพ 2 and any tree ๐‘‡ with
ฮ”(๐‘‡) โฉพ ๐‘ and characterized all trees achieving the lower
bound and the upper bound, respectively.
Theorem 141 (Lu and Xu [96], 2011). For any integer ๐‘ โฉพ 2
and any tree ๐‘‡ with ฮ”(๐‘‡) โฉพ ๐‘,
1 โฉฝ ๐‘๐‘ (๐‘‡) โฉฝ ฮ” (๐‘‡) โˆ’ ๐‘ + 1.
Then, all trees achieving the lower bound 1 can be characterized as follows.
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Let ๐‘† be a given subset of ๐‘‰(๐บ) and, for any ๐‘ฅ โˆˆ ๐‘†, let
๓ต„จ
๓ต„จ
๐‘๐‘ (๐‘ฅ, ๐‘†, ๐บ) = {๐‘ฆ โˆˆ ๐‘† โˆฉ ๐‘๐บ (๐‘ฅ) : ๓ต„จ๓ต„จ๓ต„จ๐‘๐บ (๐‘ฆ) โˆฉ ๐‘†๓ต„จ๓ต„จ๓ต„จ = ๐‘} .
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Type 2: attach a star ๐พ1,ฮ”โˆ’1 to a vertex ๐‘ฆ of ๐‘‡ by joining its
central vertex to ๐‘ฆ, where ๐‘ฆ in a ๐›พ๐‘ -set of ๐‘‡ and
๐‘‘๐‘‡ (๐‘ฆ) โฉฝ ฮ” โˆ’ 1.
Type 3: attach a double star ๐‘†(๐‘, ฮ” โˆ’ 1) to a pendant vertex ๐‘ฆ
of ๐‘‡ by coinciding its ๐‘…-central vertex with ๐‘ฆ, where
the unique neighbor of ๐‘ฆ is in a ๐›พ๐‘ -set of ๐‘‡.
Let B = {๐‘‡: a tree obtained from ๐พ1,ฮ” or ๐‘†(๐‘, ฮ”) by a
finite sequence of operations of Types 1, 2, and 3}.
Theorem 143 (Lu and Xu [96], 2011). A tree with the maximum ฮ” โฉพ ๐‘ + 1 has ๐‘-bondage number ฮ” โˆ’ ๐‘ + 1 if and only
if it belongs to B.
Theorem 144 (Lu and Xu [97], 2012). Let ๐บ๐‘š,๐‘› = ๐‘ƒ๐‘š × ๐‘ƒ๐‘› .
Then
๐‘2 (๐บ2,๐‘› ) = 1 ๐‘“๐‘œ๐‘Ÿ ๐‘› โฉพ 2,
2
๐‘2 (๐บ3,๐‘› ) = {
1
๐‘–๐‘“ ๐‘› โ‰ก 1 (mod 3) ,
๐‘œ๐‘กโ„Ž๐‘’๐‘Ÿ๐‘ค๐‘–๐‘ ๐‘’.
for ๐‘› โฉพ 2,
1
๐‘2 (๐บ4,๐‘› ) = {
2
๐‘–๐‘“ ๐‘› โ‰ก 3 (mod 4) ,
๐‘œ๐‘กโ„Ž๐‘’๐‘Ÿ๐‘ค๐‘–๐‘ ๐‘’.
for ๐‘› โฉพ 7.
(54)
Recently, Krzywkowski [98] has proposed the concept
of the nonisolating 2-bondage of a graph. The nonisolating
2-bondage number of a graph ๐บ, denoted by ๐‘2๓ธ€  (๐บ), is the
minimum cardinality among all sets of edges ๐ต โŠ† ๐ธ(๐บ) such
that ๐บ โˆ’ ๐ต has no isolated vertices and ๐›พ2 (๐บ โˆ’ ๐ต) > ๐›พ2 (๐บ). If
for every ๐ต โŠ† ๐ธ(๐บ), either ๐›พ2 (๐บ โˆ’ ๐ต) = ๐›พ2 (๐บ) or ๐บ โˆ’ ๐ต has
isolated vertices, then define ๐‘2๓ธ€  (๐บ) = 0, and say that ๐บ is a 2nonisolatedly strongly stable graph. Krzywkowski presented
some basic properties of nonisolating 2-bondage in graphs,
showed that for every nonnegative integer ๐‘˜ there exists a
tree ๐‘‡ such ๐‘2๓ธ€  (๐‘‡) = ๐‘˜, characterized all 2-non-isolatedly
strongly stable trees, and determined ๐‘2๓ธ€  (๐บ) for several special
graphs.
International Journal of Combinatorics
23
Theorem 145 (Krzywkowski [98], 2012). For any positive
integers ๐‘› and ๐‘š with ๐‘š โฉฝ ๐‘›,
{0
๐‘2๓ธ€  (๐พ๐‘› ) = { 2๐‘›
โŒŠ โŒ‹
{ 3
0
๐‘2๓ธ€  (๐‘ƒ๐‘› ) = {
1
๐‘2๓ธ€ 
๐‘–๐‘“ ๐‘› = 1, 2, 3
๓ต„จ
๓ต„จ
ฮ” ๐‘˜ (๐บ) = max {๓ต„จ๓ต„จ๓ต„จ๐‘๐‘˜ (๐‘ฅ)๓ต„จ๓ต„จ๓ต„จ : for any ๐‘ฅ โˆˆ ๐‘‰ (๐บ)} .
๐‘œ๐‘กโ„Ž๐‘’๐‘Ÿ๐‘ค๐‘–๐‘ ๐‘’,
๐‘–๐‘“ ๐‘› = 1, 2, 3
๐‘œ๐‘กโ„Ž๐‘’๐‘Ÿ๐‘ค๐‘–๐‘ ๐‘’,
0 ๐‘–๐‘“ ๐‘› = 3
{
{
(๐ถ๐‘› ) = {1 ๐‘–๐‘“ ๐‘› ๐‘–๐‘  ๐‘’V๐‘’๐‘›
{
{2 ๐‘–๐‘“ ๐‘› ๐‘–๐‘  ๐‘œ๐‘‘๐‘‘,
3
{
{
๐‘2๓ธ€  (๐พ๐‘š,๐‘› ) = {1
{
{2
For a vertex ๐‘ฅ in ๐บ, the open ๐‘˜-neighborhood ๐‘๐‘˜ (๐‘ฅ) of ๐‘ฅ is
defined as ๐‘๐‘˜ (๐‘ฅ) = {๐‘ฆ โˆˆ ๐‘‰(๐บ) : 1 โฉฝ ๐‘‘๐บ(๐‘ฅ, ๐‘ฆ) โ‰ค ๐‘˜}. The closed
๐‘˜-neighborhood ๐‘๐‘˜ [๐‘ฅ] of ๐‘ฅ in ๐บ is defined as ๐‘๐‘˜ (๐‘ฅ) โˆช {๐‘ฅ}. Let
(55)
๐‘–๐‘“ ๐‘š = ๐‘› = 3
๐‘–๐‘“ ๐‘š = ๐‘› = 4
๐‘œ๐‘กโ„Ž๐‘’๐‘Ÿ๐‘ค๐‘–๐‘ ๐‘’.
9.2. Distance Bondage Numbers. A subset ๐‘† of vertices of a
graph ๐บ is said to be a distance ๐‘˜-dominating set for ๐บ if every
vertex in ๐บ not in ๐‘† is at distance at most ๐‘˜ from some vertex
of ๐‘†. The minimum cardinality of all distance ๐‘˜-dominating
sets is called the distance ๐‘˜-domination number of ๐บ and
denoted by ๐›พ๐‘˜ (๐บ) (do not confuse with the above-mentioned
๐›พ๐‘ (๐บ)!). When ๐‘˜ = 1, a distance 1-dominating set is a normal
dominating set, and so ๐›พ1 (๐บ) = ๐›พ(๐บ) for any graph ๐บ. Thus,
the distance ๐‘˜-domination is a generalization of the classical
domination.
The relation between ๐›พ๐‘˜ and ๐›ผ๐‘˜ (the ๐‘˜-independence
number, defined in Section 2.2) for a tree obtained by Meir
and Moon [14], who proved that ๐›พ๐‘˜ (๐‘‡) = ๐›ผ2๐‘˜ (๐‘‡) for any tree
๐‘‡ (a special case for ๐‘˜ = 1 is stated in Proposition 10). The
further research results can be found in Henning et al. [99],
Tian and Xu [100โ€“103], and Liu et al. [104].
In 1998, Hartnell et al. [15] defined the distance ๐‘˜-bondage
number of ๐บ, denoted by ๐‘๐‘˜ (๐บ), to be the cardinality of a
smallest subset ๐ต of edges of ๐บ with the property that ๐›พ๐‘˜ (๐บ โˆ’
๐ต) > ๐›พ๐‘˜ (๐บ). From Theorem 6, it is clear that if ๐‘‡ is a nontrivial
tree, then 1 โฉฝ ๐‘1 (๐‘‡) โฉฝ 2. Hartnell et al. [15] generalized this
result to any integer ๐‘˜ โฉพ 1.
Theorem 146 (Hartnell et al. [15], 1998). For every nontrivial
tree ๐‘‡ and positive integer ๐‘˜, 1 โฉฝ ๐‘๐‘˜ (๐‘‡) โฉฝ 2.
Hartnell et al. [15] and Topp and Vestergaard [13] also
characterized the trees having distance ๐‘˜-bondage number 2.
In particular, the class of trees for which ๐‘1 (๐‘‡) = 2 are just
those which have a unique maximum 2-independent set (see
Theorem 11).
Since, when ๐‘˜ = 1, the distance 1-bondage number ๐‘1 (๐บ)
is the classical bondage number ๐‘(๐บ), Theorem 12 gives the
NP-hardness of deciding the distance ๐‘˜-bondage number of
general graphs.
Theorem 147. Given a nonempty undirected graph ๐บ and
positive integers ๐‘˜ and ๐‘ with ๐‘ โฉฝ ๐œ€(๐บ), determining wether
or not ๐‘๐‘˜ (๐บ) โฉฝ ๐‘ is NP-hard.
(56)
Clearly, ฮ” 1 (๐บ) = ฮ”(๐บ). The ๐‘˜th power of a graph ๐บ is the
graph ๐บ๐‘˜ with vertex-set ๐‘‰(๐บ๐‘˜ ) = ๐‘‰(๐บ) and edge-set ๐ธ(๐บ๐‘˜ ) =
{๐‘ฅ๐‘ฆ : 1 โฉฝ ๐‘‘๐บ(๐‘ฅ, ๐‘ฆ) โฉฝ ๐‘˜}. The following lemma holds directly
from the definition of ๐บ๐‘˜ .
Proposition 148 (Tian and Xu [101]). ฮ”(๐บ๐‘˜ ) = ฮ” ๐‘˜ (๐บ) and
๐›พ(๐บ๐‘˜ ) = ๐›พ๐‘˜ (๐บ) for any graph ๐บ and each ๐‘˜ โฉพ 1.
A graph ๐บ is ๐‘˜-distance domination-critical, or ๐›พ๐‘˜ -critical
for short, if ๐›พ๐‘˜ (๐บ โˆ’ ๐‘ฅ) < ๐›พ๐‘˜ (๐บ) for every vertex ๐‘ฅ in ๐บ,
proposed by Henning et al. [105]
Proposition 149 (Tian and Xu [101]). For each ๐‘˜ โฉพ 1, a graph
๐บ is ๐›พ๐‘˜ -critical if and only if ๐บ๐‘˜ is ๐›พ๐‘˜ -critical.
By the above facts, we suggest the following worthwhile
problem.
Problem 6. Can we generalize the results on the bondage for
๐บ to ๐บ๐‘˜ ? In particular, do the following propositions hold?
(a) ๐‘(๐บ๐‘˜ ) = ๐‘๐‘˜ (๐บ) for any graph ๐บ and each ๐‘˜ โฉพ 1.
(b) ๐‘(๐บ๐‘˜ ) โฉฝ ฮ” ๐‘˜ (๐บ) if ๐บ is not ๐›พ๐‘˜ -critical.
Let ๐‘˜ and ๐‘ be positive integers. A subset ๐‘† of ๐‘‰(๐บ) is
defined to be a (๐‘˜, ๐‘)-dominating set of ๐บ if, for any vertex
๐‘ฅ โˆˆ ๐‘†, |๐‘๐‘˜ (๐‘ฅ) โˆฉ ๐‘†| โฉพ ๐‘. The (๐‘˜, ๐‘)-domination number of
๐บ, denoted by ๐›พ๐‘˜,๐‘ (๐บ), is the minimum cardinality among
all (๐‘˜, ๐‘)-dominating sets of ๐บ. Clearly, for a graph ๐บ, a
(1, 1)-dominating set is a classical dominating set, a (๐‘˜, 1)dominating set is a distance ๐‘˜-dominating set, and a (1, ๐‘)dominating set is the above-mentioned ๐‘-dominating set.
That is, ๐›พ1,1 (๐บ) = ๐›พ(๐บ), ๐›พ๐‘˜,1 (๐บ) = ๐›พ๐‘˜ (๐บ), and ๐›พ1,๐‘ (๐บ) = ๐›พ๐‘ (๐บ).
The concept of (๐‘˜, ๐‘)-domination in a graph ๐บ is a generalized domination which combines distance ๐‘˜-domination
and ๐‘-domination in ๐บ. So, the investigation of (๐‘˜, ๐‘)domination of ๐บ is more interesting and has received the
attention of many researchers; see, for example, [106โ€“109].
More general, Li and Xu [110] proposed the concept of the
(โ„“, ๐‘ค)-domination number of a ๐‘ค-connected graph. A subset
๐‘† of ๐‘‰(๐บ) is defined to be an (โ„“, ๐‘ค)-dominating set of ๐บ if,
for any vertex ๐‘ฅ โˆˆ ๐‘†, there are ๐‘ค internally disjoint paths
of length at most โ„“ from ๐‘ฅ to some vertex in ๐‘†. The (โ„“, ๐‘ค)domination number of ๐บ, denoted by ๐›พโ„“,๐‘ค (๐บ), is the minimum
cardinality among all (โ„“, ๐‘ค)-dominating sets of ๐บ. Clearly,
๐›พ๐‘˜,1 (๐บ) = ๐›พ๐‘˜ (๐บ) and ๐›พ1,๐‘ (๐บ) = ๐›พ๐‘ (๐บ).
It is quite natural to propose the concept of bondage number for (๐‘˜, ๐‘)-domination or (โ„“, ๐‘ค)-domination. However, as
far as we know, none has proposed this concept until today.
This is a worthwhile topic for further research.
24
International Journal of Combinatorics
9.3. Fractional Bondage Numbers. If ๐œŽ is a function mapping
the vertex-set ๐‘‰ into some set of real numbers, then for any
subset ๐‘† โŠ† ๐‘‰; let ๐œŽ(๐‘†) = โˆ‘๐‘ฅโˆˆ๐‘† ๐œŽ(๐‘ฅ). Also let |๐œŽ| = ๐œŽ(๐‘‰).
A real-value function ๐œŽ : ๐‘‰ โ†’ [0, 1] is a dominating
function of a graph ๐บ if for every ๐‘ฅ โˆˆ ๐‘‰(๐บ), ๐œŽ(๐‘๐บ[๐‘ฅ]) โฉพ 1.
Thus, if ๐‘† is a dominating set of ๐บ, ๐œŽ is a function, where
1
๐œŽ (๐‘ฅ) = {
0
if ๐‘ฅ โˆˆ ๐‘†,
otherwise,
๐‘“ (๐บ) = โˆ‘ ๐‘“ (๐‘ข) .
(57)
then ๐œŽ is a dominating function of ๐บ. The fractional domination number of ๐บ, denoted by ๐›พ๐น (๐บ), is defined as follows.
๐›พ๐น (๐บ) = min {|๐œŽ| : ๐œŽ is a domination function of ๐บ} .
(58)
The ๐›พ๐น -bondage number of ๐บ, denoted by ๐‘๐น (๐บ), is
defined as the minimum cardinality of a subset ๐ต โŠ† ๐ธ whose
removal results in ๐›พ๐น (๐บ โˆ’ ๐ต) > ๐›พ๐น (๐บ).
Hedetniemi et al. [111] are the first to study fractional
domination although Farber [112] introduced the idea indirectly. The concept of the fractional domination number
was proposed by Domke and Laskar [113] in 1997. The
fractional domination numbers for some ordinary graphs are
determined.
Proposition 150 (Domke et al. [114], 1998; Domke and Laskar
[113], 1997). (a) If ๐บ is a ๐‘˜-regular graph with order ๐‘›, then
๐›พ๐น (๐บ) = ๐‘›/๐‘˜ + 1;
(b) ๐›พ๐น (๐ถ๐‘› ) = ๐‘›/3, ๐›พ๐น (๐‘ƒ๐‘› ) = โŒˆ๐‘›/3โŒ‰, ๐›พ๐น (๐พ๐‘› ) = 1;
(c) ๐›พ๐น (๐บ) = 1 if and only if ฮ”(๐บ) = ๐‘› โˆ’ 1;
(d) ๐›พ๐น (๐‘‡) = ๐›พ(๐‘‡) for any tree ๐‘‡;
(e) ๐›พ๐น (๐พ๐‘›,๐‘š ) = (๐‘›(๐‘š + 1) + ๐‘š(๐‘› + 1))/(๐‘›๐‘š โˆ’ 1), where
max{๐‘›, ๐‘š} > 1.
The assertions (a)โ€“(d) are due to Domke et al. [114] and
the assertion (e) is due to Domke and Laskar [113]. According
to these results, Domke and Laskar [113] determined the ๐›พ๐น bondage numbers for these graphs.
Theorem 151 (Domke and Laskar [113], 1997). ๐‘๐น (๐พ๐‘› ) =
โŒˆ๐‘›/2โŒ‰; ๐‘๐น (๐พ๐‘›,๐‘š ) = 1 where max{๐‘›, ๐‘š} > 1,
2 ๐‘–๐‘“ ๐‘› โ‰ก 0 (mod 3) ,
๐‘๐น (๐ถ๐‘› ) = {
1 ๐‘œ๐‘กโ„Ž๐‘’๐‘Ÿ๐‘ค๐‘–๐‘ ๐‘’,
2 ๐‘–๐‘“ ๐‘› โ‰ก 1 (mod 3) ,
๐‘๐น (๐‘ƒ๐‘› ) = {
1 ๐‘œ๐‘กโ„Ž๐‘’๐‘Ÿ๐‘ค๐‘–๐‘ ๐‘’,
(59)
๐‘๐น (๐‘‡) = min {๐ต โŠ† ๐ธ (๐‘‡) : ๐›พ๐น (๐‘‡ โˆ’ ๐ต) > ๐›พ๐น (๐‘‡)}
(61)
The minimum weight of a Roman dominating function on
a graph ๐บ is called the Roman domination number of ๐บ,
denoted by ๐›พ RM (๐บ).
A Roman dominating function ๐‘“ : ๐‘‰ โ†’ {0, 1, 2}
can be represented by the ordered partition (๐‘‰0 , ๐‘‰1 , ๐‘‰2 ) (or
๐‘“
๐‘“
๐‘“
(๐‘‰0 , ๐‘‰1 , ๐‘‰2 ) to refer to ๐‘“) of ๐‘‰, where ๐‘‰๐‘– = {V โˆˆ ๐‘‰ | ๐‘“(V) = ๐‘–}.
In this representation, its weight ๐‘“(๐บ) = |๐‘‰1 | + 2|๐‘‰2 |. It is
๐‘“
๐‘“
clear that ๐‘‰1 โˆช ๐‘‰2 is a dominating set of ๐บ, called the Roman
๐‘“
๐‘“
๐‘“
dominating set, denoted by ๐ทR = (๐‘‰1 , ๐‘‰2 ). Since ๐‘‰1 โˆช ๐‘‰2 is
a dominating set when ๐‘“ is a Roman dominating function,
and since placing weight 2 at the vertices of a dominating set
yields a Roman dominating function, in [115], it is observed
that
๐›พ (๐บ) โฉฝ ๐›พ RM (๐บ) โฉฝ 2๐›พ (๐บ) .
(62)
A graph ๐บ is called to be Roman if ๐›พ RM (๐บ) = 2๐›พ(๐บ).
The definition of the Roman domination function is
proposed implicitly by Stewart [116] and ReVelle and Rosing
[117]. Roman dominating numbers have been deeply studied.
In particular, Bahremandpour et al. [118] showed that the
problem determining the Roman domination number is NPcomplete even for bipartite graphs.
Let ๐บ be a graph with maximum degree at least two.
The Roman bondage number ๐‘ RM (๐บ) of ๐บ is the minimum
cardinality of all sets ๐ธ๓ธ€  โŠ† ๐ธ for which ๐›พ RM (๐บ โˆ’ ๐ธ๓ธ€  ) >
๐›พ RM (๐บ). Since in study of Roman bondage number the
assumption ฮ”(๐บ) โฉพ 2 is necessary, we always assume that
when we discuss ๐‘ RM (๐บ), all graphs involved satisfy ฮ”(๐บ) โฉพ
2. The Roman bondage number ๐‘ RM (๐บ) is introduced by
Jafari Rad and Volkmann in [119].
Recently, Bahremandpour et al. [118] have shown that the
problem determining the Roman bondage number is NPhard even for bipartite graphs.
Roman Bondage Problem
Instance: a nonempty bipartite graph ๐บ and a positive
integer ๐‘˜.
Question: is ๐‘ RM (๐บ) โฉฝ ๐‘˜?
(60)
Theorem 152 (Bahremandpour et al. [118], 2013). The Roman
bondage problem is NP-hard even for bipartite graphs.
Except for these, no results on ๐‘๐น (๐บ) have been known as
The exact value of ๐‘RM (๐บ) is known only for a few family
of graphs including the complete graphs, cycles, and paths.
= ๐‘ (๐‘‡) โฉฝ 2.
yet.
๐‘ขโˆˆ๐‘‰(๐บ)
Problem 7. Consider the decision problem:
It is easy to see that for any tree ๐‘‡, ๐‘๐น (๐‘‡) โฉฝ 2. In fact, since
๐›พ๐น (๐‘‡) = ๐›พ(๐‘‡) for any tree ๐‘‡, ๐›พ๐น (๐‘‡๓ธ€  ) = ๐›พ(๐‘‡๓ธ€  ) for any subgraph
๐‘‡๓ธ€  of ๐‘‡. It follows that
= min {๐ต โŠ† ๐ธ (๐‘‡) : ๐›พ (๐‘‡ โˆ’ ๐ต) > ๐›พ (๐‘‡)}
9.4. Roman Bondage Numbers. A Roman dominating function on a graph ๐บ is a labeling ๐‘“ : ๐‘‰ โ†’ {0, 1, 2} such that
every vertex with label 0 has at least one neighbor with label
2. The weight of a Roman dominating function ๐‘“ on ๐บ is the
value
International Journal of Combinatorics
25
Theorem 153 (Jafari Rad and Volkmann [119], 2011). For any
positive integers ๐‘› and ๐‘š with ๐‘š โฉฝ ๐‘›,
2,
๐‘RM (๐‘ƒ๐‘› ) = {
1,
๐‘–๐‘“ ๐‘› โ‰ก 2 (mod 3) ,
๐‘œ๐‘กโ„Ž๐‘’๐‘Ÿ๐‘ค๐‘–๐‘ ๐‘’,
3,
๐‘RM (๐ถ๐‘› ) = {
2,
๐‘–๐‘“ ๐‘› = 2 (mod 3) ,
๐‘œ๐‘กโ„Ž๐‘’๐‘Ÿ๐‘ค๐‘–๐‘ ๐‘’,
๐‘›
๐‘RM (๐พ๐‘› ) = โŒˆ โŒ‰
2
1
{
{
๐‘RM (๐พ๐‘š,๐‘› ) = {4
{
{๐‘š
Prove or Disprove: for any connected graph ๐บ of order ๐‘› โ‰ฅ 3,
(i) ๐‘RM (๐บ) = ๐‘› โˆ’ 1 if and only if ๐บ โ‰… ๐พ3 ; (ii) ๐‘RM (๐บ) โฉฝ ๐‘› โˆ’
๐›พ๐‘… (๐บ) + 1.
(63)
๐‘“๐‘œ๐‘Ÿ ๐‘› โฉพ 3,
๐‘–๐‘“ ๐‘š = 1 ๐‘Ž๐‘›๐‘‘ ๐‘› =ฬธ 1,
๐‘–๐‘“ ๐‘š = ๐‘› = 3,
๐‘œ๐‘กโ„Ž๐‘’๐‘Ÿ๐‘ค๐‘–๐‘ ๐‘’.
Using Lemma 154, the third conclusion in Theorem 153
can be generalized to more general case, which is similar to
Lemma 49.
Theorem 155 (Jafari Rad and Volkmann [119], 2011). Let ๐บ be
a graph with order ๐‘› โ‰ฅ 3 and ๐‘ก the number of vertices of degree
๐‘› โˆ’ 1 in ๐บ. If ๐‘ก โฉพ 1, then ๐‘RM (๐บ) = โŒˆ๐‘ก/2โŒ‰.
Ebadi and PushpaLatha [120] conjectured that ๐‘RM (๐บ) โฉฝ
๐‘› โˆ’ 1 for any graph ๐บ of order ๐‘› โฉพ 3. Dehgardi et al.
[121], Akbari and Qajar [122], independently, showed that this
conjecture is true.
Theorem 156. ๐‘RM (๐บ) โฉฝ ๐‘› โˆ’ 1 for any connected graph ๐บ of
order ๐‘› โฉพ 3.
For a complete ๐‘ก-partite graph, we have the following
result.
Theorem 157 (Hu and Xu [123], 2011). Let ๐บ = ๐พ๐‘š1 ,๐‘š2 ,...,๐‘š๐‘ก be
a complete ๐‘ก-partite graph with ๐‘š1 = โ‹… โ‹… โ‹… = ๐‘š๐‘– < ๐‘š๐‘–+1 โ‰ค โ‹… โ‹… โ‹… โฉฝ
๐‘š๐‘ก , ๐‘ก โฉพ 2 and ๐‘› = โˆ‘๐‘ก๐‘—=1 ๐‘š๐‘— . Then
๐‘–๐‘“ ๐‘š๐‘– = 1 ๐‘Ž๐‘›๐‘‘ ๐‘› โฉพ 3,
๐‘–๐‘“
๐‘–๐‘“
๐‘–๐‘“
๐‘–๐‘“
๐‘–๐‘“
๐‘š๐‘–
๐‘š๐‘–
๐‘š๐‘–
๐‘š๐‘–
๐‘š๐‘–
=2
=2
=3
=3
โฉพ3
๐‘Ž๐‘›๐‘‘
๐‘Ž๐‘›๐‘‘
๐‘Ž๐‘›๐‘‘
๐‘Ž๐‘›๐‘‘
๐‘Ž๐‘›๐‘‘
๐‘– = 1,
๐‘– โฉพ 2,
๐‘– = ๐‘ก = 2,
๐‘– = ๐‘ก โฉพ 3,
๐‘š๐‘ก โฉพ 4.
Note that a complete ๐‘ก-partite graph ๐พ๐‘ก (3) is (๐‘› โˆ’ 3)regular and ๐‘RM (๐พ๐‘ก (3)) = ๐‘› โˆ’ 1 for ๐‘ก โฉพ 3, where ๐‘› = 3๐‘ก.
Hu and Xu further determined that ๐‘๐‘… (๐บ) = ๐‘› โˆ’ 2 for any
(๐‘› โˆ’ 3)-regular graph ๐บ of order ๐‘› โฉฝ 5 except ๐พ๐‘ก (3).
Theorem 158 (Hu and Xu [123], 2011). For any (๐‘› โˆ’ 3)-regular
graph ๐บ of order ๐‘› other than ๐พ๐‘ก (3), ๐‘RM (๐บ) = ๐‘› โˆ’ 2 for ๐‘› โฉพ 5.
For a tree ๐‘‡ with order ๐‘› โฉพ 3, Ebadi and PushpaLatha
[120], and Jafari Rad and Volkmann [119], independently,
obtained an upper bound on ๐‘RM (๐‘‡).
Lemma 154 (Cockayne et al. [115], 2004). If ๐บ is a graph of
order ๐‘› and contains vertices of degree ๐‘› โˆ’ 1, then ๐›พRM (๐บ) = 2.
๐‘–
{
โŒˆ โŒ‰
{
{
2
{
{
{
{2
{
{
{
๐‘RM (๐บ) = {๐‘–
{
{
4
{
{
{
{
{
๐‘›โˆ’1
{
{
{๐‘› โˆ’ ๐‘š๐‘ก
a negative answer to the following two problems posed by
Dehgardi et al. [121].
(64)
Consider a complete ๐‘ก-partite graph ๐พ๐‘ก (3) for ๐‘ก โฉพ 3,
๐‘RM (๐พ๐‘ก (3)) = ๐‘› โˆ’ 1 by Theorem 157, where ๐‘› = 3๐‘ก,
which shows that this upper bound of ๐‘› โˆ’ 1 on ๐‘RM (๐บ) in
Theorem 156 is sharp. This fact and ๐›พ๐‘… (๐พ๐‘ก (3)) = 4 give
Theorem 159. ๐‘RM (๐‘‡) โฉฝ 3 for any tree ๐‘‡ with order ๐‘› โฉพ 3.
Theorem 160 (Bahremandpour et al. [118], 2013). ๐‘RM (๐‘ƒ2 ×
๐‘ƒ๐‘› ) = 2 for ๐‘› โฉพ 2.
Theorem 161 (Jafari Rad and Volkmann [119], 2011). Let ๐บ be
a graph with order at least three.
(a) If (๐‘ฅ, ๐‘ฆ, ๐‘ง) is a path of length 2 in ๐บ, then ๐‘RM (๐บ) โฉฝ
๐‘‘๐บ(๐‘ฅ) + ๐‘‘๐บ(๐‘ฆ) + ๐‘‘๐บ(๐‘ง) โˆ’ 3 โˆ’ |๐‘๐บ(๐‘ฅ) โˆฉ ๐‘๐บ(๐‘ฆ)|.
(b) If ๐บ is connected, then ๐‘RM (๐บ) โฉฝ ๐œ†(๐บ) + 2ฮ”(๐บ) โˆ’ 3.
Theorem 161 (b) implies ๐‘RM (๐บ) โฉฝ ๐›ฟ(๐บ) + 2ฮ”(๐บ) โˆ’ 3 for a
connected graph ๐บ. Note that for a planar graph ๐บ, ๐›ฟ(๐บ) โฉฝ 5;
moreover, ๐›ฟ(๐บ) โฉฝ 3 if the girth at least 4, and ๐›ฟ(๐บ) โฉฝ 2 if the
girth at least 6. These two facts show that ๐‘RM (๐บ) โฉฝ 2ฮ”(๐บ) +
2 for a connected planar graph ๐บ. Jafari Rad and Volkmann
[124] improved this bound.
Theorem 162 (Jafari Rad and Volkmann [124], 2011). For any
connected planar graph ๐บ of order ๐‘› โฉพ 3 with girth ๐‘”(๐บ),
๐‘RM (๐บ)
2ฮ” (๐บ) ,
{
{
{
{
ฮ” (๐บ) + 6,
{
{
{
{
{
ฮ” (๐บ) + 5
{
{
โฉฝ {ฮ” (๐บ) + 4
{
{
{
ฮ” (๐บ) + 3
{
{
{
{
{
{
{ฮ” (๐บ) + 2
{ฮ” (๐บ) + 1
๐‘–๐‘“
๐‘–๐‘“
๐‘–๐‘“
๐‘–๐‘“
๐‘–๐‘“
๐บ ๐‘๐‘œ๐‘›๐‘ก๐‘Ž๐‘–๐‘›๐‘  ๐‘›๐‘œ V๐‘’๐‘Ÿ๐‘ก๐‘–๐‘๐‘’๐‘  ๐‘œ๐‘“ ๐‘‘๐‘’๐‘”๐‘Ÿ๐‘’๐‘’ ๐‘“๐‘–V๐‘’,
๐‘” (๐บ) โฉพ 4,
๐‘” (๐บ) โฉพ 5,
๐‘” (๐บ) โฉพ 6,
๐‘” (๐บ) โฉพ 8.
(65)
According to Theorem 157, ๐‘RM (๐ถ๐‘› ) = 3 = ฮ”(๐ถ๐‘› )+1 for a
cycle ๐ถ๐‘› of length ๐‘› โฉพ 8 with ๐‘› โ‰ก 2 (mod 3), and therefore
the last upper bound in Theorem 162 is sharp, at least for ฮ” =
2.
Combining the fact that every planar graph ๐บ with
minimum degree 5 contains an edge ๐‘ฅ๐‘ฆ with ๐‘‘๐บ(๐‘ฅ) = 5
and ๐‘‘๐บ(๐‘ฆ) โˆˆ {5, 6} with Theorem 161 (a), Akbari et al. [125]
obtained the following result.
26
International Journal of Combinatorics
···
Theorem 167 (Bahremandpour et al. [118], 2013). For every
Roman graph ๐บ,
···
๐‘RM (๐บ) โฉพ ๐‘ (๐บ) .
The bound is sharp for cycles on ๐‘› vertices where ๐‘› โ‰ก 0 (mod
3).
๐บ
···
(67)
The strict inequality in Theorem 167 can hold, for example, ๐‘(๐ถ3๐‘˜+2 ) = 2 < 3 = ๐‘RM (๐ถ3๐‘˜+2 ) by Theorem 157.
A graph ๐บ is called to be vertex Roman dominationcritical if ๐›พRM (๐บ โˆ’ ๐‘ฅ) < ๐›พRM (๐บ) for every vertex ๐‘ฅ in ๐บ. If
๐บ has no isolated vertices, then ๐›พRM (๐บ) โฉฝ 2๐›พ(๐บ) โฉฝ 2๐›ฝ(๐บ).
If ๐›พRM (๐บ) = 2๐›ฝ(๐บ), then ๐›พRM (๐บ) = 2๐›พ(๐บ) and hence ๐บ is a
Roman graph. In [30], Volkmann gave a lot of graphs with
๐›พ(๐บ) = ๐›ฝ(๐บ). The following result is similar to Theorem 57.
···
ฬ‚ is constructed from ๐บ.
Figure 9: The graph ๐บ
Theorem 163 (Akbari et al. [125], 2012). ๐‘RM (๐บ) โฉฝ 15 for
every planar graph ๐บ.
It remains open to show whether the bound in
Theorem 163 is sharp or not. Though finding a planar
graph ๐บ with ๐‘RM (๐บ) = 15 seems to be difficult, Akbari et
al. [125] constructed an infinite family of planar graphs with
Roman bondage number equal to 7 by proving the following
result.
Theorem 164 (Akbari et al. [125], 2012). Let ๐บ be a graph of
ฬ‚ is the graph of order 5๐‘› obtained from ๐บ by
order ๐‘› and ๐บ
attaching the central vertex of a copy of a path ๐‘ƒ5 to each vertex
ฬ‚ = 4๐‘› and ๐‘RM (๐บ)
ฬ‚ = ๐›ฟ(๐บ) + 2.
of ๐บ (see Figure 9). Then ๐›พRM (๐บ)
By Theorem 164, infinite many planar graphs with Roman
bondage number 7 can be obtained by considering any planar
graph ๐บ with ๐›ฟ(๐บ) = 5 (e.g., the icosahedron graph).
Conjecture 165 (Akbari et al. [125], 2012). The Roman
bondage number of every planar graph is at most 7.
For general bounds, the following observation is directly
obtained, which is similar to Observation 1.
Observation 4. Let ๐บ be a graph of order ๐‘› with maximum
degree at least two. Assume that ๐ป is a spanning subgraph
obtained from ๐บ by removing ๐‘˜ edges. If ๐›พRM (๐ป) = ๐›พRM (๐บ),
then ๐‘๐‘… (๐ป) โฉฝ ๐‘RM (๐บ) โฉฝ ๐‘RM (๐ป) + ๐‘˜.
Theorem 166 (Bahremandpour et al. [118], 2013). For any
connected graph ๐บ of order ๐‘›, if ๐‘› โฉพ 3 and ๐›พRM (๐บ) = ๐›พ(๐บ) + 1,
then
Theorem 168 (Bahremandpour et al. [118], 2013). For any
graph ๐บ, if ๐›พRM (๐บ) = 2๐›ฝ(๐บ), then
(a) ๐‘RM (๐บ) โฉพ ๐›ฟ(๐บ);
(b) ๐‘RM (๐บ) โฉพ ๐›ฟ(๐บ) + 1 if ๐บ is a vertex Roman dominationcritical graph.
If ๐›พRM (๐บ) = 2, then obviously ๐‘RM (๐บ) โฉฝ ๐›ฟ(๐บ). So we
assume ๐›พRM (๐บ) โฉพ 3. Dehgardi et al. [121] proved ๐‘RM (๐บ) โฉฝ
(๐›พRM (๐บ) โˆ’ 2)ฮ”(๐บ) + 1 and posed the following problem: if ๐บ
is a connected graph of order ๐‘› โฉพ 4 with ๐›พRM (๐บ) โฉพ 3, then
๐‘RM (๐บ) โฉฝ (๐›พRM (๐บ) โˆ’ 2) ฮ” (๐บ) .
(68)
Theorem 161 (a) shows that the inequality (68) holds if
๐›พRM (๐บ) โฉพ 5. Thus, the bound in (68) is of interest only when
๐›พRM (๐บ) is 3 or 4.
Lemma 169 (Bahremandpour et al. [118], 2013). For any
nonempty graph ๐บ of order ๐‘› โฉพ 3, ๐›พRM (๐บ) = 3 if and only
if ฮ”(๐บ) = ๐‘› โˆ’ 2.
The following result shows that (68) holds for all graphs
๐บ of order ๐‘› โฉพ 4 with ๐›พRM (๐บ) = 3 or 4, which improves
Theorem 161 (b).
Theorem 170 (Bahremandpour et al. [118], 2013). For any
connected graph ๐บ of order ๐‘› โฉพ 4,
๐‘RM (๐บ) โ‰ค {
ฮ” (๐บ) = ๐‘› โˆ’ 2
ฮ” (๐บ) + ๐›ฟ (๐บ) โˆ’ 1
๐‘–๐‘“ ๐›พRM (๐บ) = 3,
๐‘–๐‘“ ๐›พRM (๐บ) = 4
(69)
with the first equality if and only if ๐บ โ‰… ๐ถ4 .
๐‘RM (๐บ) โฉฝ min {๐‘ (๐บ) , ๐‘›ฮ” } ,
(66)
where ๐‘›ฮ” is the number of vertices with maximum degree ฮ” in
๐บ.
Observation 5. For any graph ๐บ, if ๐›พ(๐บ) = ๐›พRM (๐บ), then
๐‘RM (๐บ) โฉฝ ๐‘(๐บ).
Recently, Akbari and Qajar [122] have proved the following result.
Theorem 171 (Akbari and Qajar [122]). For any connected
graph ๐บ of order ๐‘› โฉพ 3,
๐‘RM (๐บ) โฉฝ ๐‘› โˆ’ ๐›พRM (๐บ) + 5.
(70)
International Journal of Combinatorics
27
We conclude this section with the following problems.
Theorem 172 (Dehgardi et al. [128]). For any integer ๐‘› โฉพ 3,
Problem 8. Characterize all connected graphs ๐บ of order ๐‘› โฉพ
3, for which ๐‘RM (๐บ) = ๐‘› โˆ’ 1.
๐‘๐‘Ÿ2 (๐‘ƒ๐‘› ) = 1 ๐‘“๐‘œ๐‘Ÿ ๐‘› โฉพ 2,
Problem 9. Prove or disprove: if ๐บ is a connected graph of
order ๐‘› โฉพ 3, then
๐‘๐‘Ÿ2 (๐ถ๐‘› ) = {
๐‘RM (๐บ) โฉฝ ๐‘› โˆ’ ๐›พRM (๐บ) + 3.
(71)
Note that ๐›พ๐‘… (๐พ๐‘ก (3)) = 4 and ๐‘RM (๐พ๐‘ก (3)) = ๐‘›โˆ’1, where ๐‘› =
3๐‘ก for ๐‘ก โฉพ 3. The upper bound on ๐‘RM (๐บ) given in Problem 9
is sharp if Problem 9 has positive answer.
9.5. Rainbow Bondage Numbers. For a positive integer ๐‘˜, a
๐‘˜-rainbow dominating function of a graph ๐บ is a function ๐‘“
from the vertex-set ๐‘‰(๐บ) to the set of all subsets of the set
{1, 2, . . . , ๐‘˜} such that for any vertex ๐‘ฅ in ๐บ with ๐‘“(๐‘ฅ) = 0 the
condition
โ‹ƒ ๐‘“ (๐‘ฆ) = {1, 2, . . . , ๐‘˜}
๐‘ฆโˆˆ๐‘๐บ (๐‘ฅ)
(72)
is fulfilled. The weight of a ๐‘˜-rainbow dominating function ๐‘“
on ๐บ is the value
๓ต„จ
๓ต„จ
๐‘“ (๐บ) = โˆ‘ ๓ต„จ๓ต„จ๓ต„จ๐‘“ (๐‘ฅ)๓ต„จ๓ต„จ๓ต„จ .
๐‘ฅโˆˆ๐‘‰(๐บ)
(73)
The ๐‘˜-rainbow domination number of a graph ๐บ, denoted by
๐›พ๐‘Ÿ๐‘˜ (๐บ), is the minimum weight of a ๐‘˜-rainbow dominating
function ๐‘“ of ๐บ.
Note that ๐›พ๐‘Ÿ1 (๐บ) is the classical domination number ๐›พ(๐บ).
The ๐‘˜-rainbow domination number is introduced by BresฬŒar
et al. [126]. Rainbow domination of a graph ๐บ coincides with
ordinary domination of the Cartesian product of ๐บ with the
complete graph, in particular, ๐›พ๐‘Ÿ๐‘˜ (๐บ) = ๐›พ(๐บ × ๐พ๐‘˜ ) for any
positive integer ๐‘˜ and any graph ๐บ [126]. Hartnell and Rall
[127] proved that ๐›พ(๐‘‡) < ๐›พ๐‘Ÿ2 (๐‘‡) for any tree ๐‘‡, no graph has
๐›พ = ๐›พ๐‘Ÿ๐‘˜ for ๐‘˜ โฉพ 3, for any ๐‘˜ โฉพ 2, and any graph ๐บ of order ๐‘›:
min {๐‘›, ๐›พ (๐บ) + ๐‘˜ โˆ’ 2} โฉฝ ๐›พ๐‘Ÿ๐‘˜ (๐บ) โฉฝ ๐‘˜๐›พ (๐บ) .
(74)
The introduction of rainbow domination is motivated
by the study of paired domination in Cartesian products
of graphs, where certain upper bounds can be expressed in
terms of rainbow domination; that is, for any graph ๐บ and
any graph ๐ป, if ๐‘‰(๐ป) can be partitioned into ๐‘˜๐›พ๐‘ƒ -sets, then
๐›พ๐‘ƒ (๐บ × ๐ป) โฉฝ (1/๐‘˜)๐œ(๐ป)๐›พ๐‘Ÿ๐‘˜ (๐บ), proved by BresฬŒar et al. [126].
Dehgardi et al. [128] introduced the concept of ๐‘˜-rainbow
bondage number of graphs. Let ๐บ be a graph with maximum
degree at least two. The ๐‘˜-rainbow bondage number ๐‘๐‘Ÿ๐‘˜ (๐บ)
of ๐บ is the minimum cardinality of all sets ๐ต โŠ† ๐ธ(๐บ) for
which ๐›พ๐‘Ÿ๐‘˜ (๐บ โˆ’ ๐ต) > ๐›พ๐‘Ÿ๐‘˜ (๐บ). Since in the study of ๐‘˜-rainbow
bondage number the assumption ฮ”(๐บ) โฉพ 2 is necessary,
we always assume that when we discuss ๐‘๐‘Ÿ๐‘˜ (๐บ), all graphs
involved satisfy ฮ”(๐บ) โฉพ 2.
The 2-rainbow bondage number of some special families
of graphs has been determined.
1 ๐‘–๐‘“ ๐‘› โ‰ก 0 (mod 4) ,
2 ๐‘œ๐‘กโ„Ž๐‘’๐‘Ÿ๐‘ค๐‘–๐‘ ๐‘’,
1
{
{
{
{3
๐‘๐‘Ÿ2 (๐พ๐‘›,๐‘› ) = {
{
6
{
{
{๐‘š
๐‘–๐‘“
๐‘–๐‘“
๐‘–๐‘“
๐‘–๐‘“
๐‘› = 2,
๐‘› = 3,
๐‘› = 4,
๐‘› โฉพ 5.
(75)
Theorem 173 (Dehgardi et al. [128]). For any connected graph
๐บ of order ๐‘› โฉพ 3, ๐‘๐‘Ÿ2 (๐บ) โฉฝ ๐œ†(๐บ) + 2ฮ”(๐บ) โˆ’ 3.
Theorem 174 (Dehgardi et al. [128]). For any tree ๐‘‡ of order
๐‘› โฉพ 3, ๐‘๐‘Ÿ2 (๐‘‡) โฉฝ 2.
Theorem 175 (Dehgardi et al. [128]). If ๐บ is a planar graph
with maximum degree at least two, then ๐‘๐‘Ÿ2 (๐บ) โฉฝ 15. Moreover,
if the girth ๐‘”(๐บ) โฉพ 4, then ๐‘๐‘Ÿ2 (๐บ) โฉฝ 11 and if ๐‘”(๐บ) โฉพ 6, then
๐‘๐‘Ÿ2 (๐บ) โฉฝ 9.
Theorem 176 (Dehgardi et al. [128]). For a complete bipartite
graph ๐พ๐‘,๐‘ž , if 2๐‘˜ + 1 โฉฝ ๐‘ โฉฝ ๐‘ž, then ๐‘๐‘Ÿ๐‘˜ (๐พ๐‘,๐‘ž ) = ๐‘.
Theorem 177 (Dehgardi et al. [128]). For a complete graph ๐พ๐‘› ,
if ๐‘› โฉพ ๐‘˜ + 1, then ๐‘๐‘Ÿ๐‘˜ (๐พ๐‘› ) = โŒˆ๐‘˜๐‘›/(๐‘˜ + 1)โŒ‰.
The special case ๐‘˜ = 1 of Theorem 177 can be found in
Theorem 1. The following is a simple observation similar to
Observation 1.
Observation 6 (Dehgardi et al. [128]). Let ๐บ be a graph
with maximum degree at least two, ๐ป a spanning subgraph
obtained from ๐บ by removing ๐‘˜ edges. If ๐›พ๐‘Ÿ๐‘˜ (๐ป) = ๐›พ๐‘Ÿ๐‘˜ (๐บ),
then ๐‘๐‘Ÿ๐‘˜ (๐ป) โฉฝ ๐‘๐‘Ÿ๐‘˜ (๐บ) โฉฝ ๐‘๐‘Ÿ๐‘˜ (๐ป) + ๐‘˜.
Theorem 178 (Dehgardi et al. [128]). For every graph ๐บ with
๐›พ๐‘Ÿ๐‘˜ (๐บ) = ๐‘˜๐›พ(๐บ), ๐‘๐‘Ÿ๐‘˜ (๐บ) โฉพ ๐‘(๐บ).
A graph ๐บ is said to be vertex ๐‘˜-rainbow dominationcritical if ๐›พ๐‘Ÿ๐‘˜ (๐บ โˆ’ ๐‘ฅ) < ๐›พ๐‘Ÿ๐‘˜ (๐บ) for every vertex ๐‘ฅ in ๐บ. It is
clear that if ๐บ has no isolated vertices, then ๐›พ๐‘Ÿ๐‘˜ (๐บ) โฉฝ ๐‘˜๐›พ(๐บ) โฉฝ
๐‘˜๐›ฝ(๐บ), where ๐›ฝ(๐บ) is the vertex-covering number of ๐บ. Thus,
if ๐›พ๐‘Ÿ๐‘˜ (๐บ) = ๐‘˜๐›ฝ(๐บ), then ๐›พ๐‘Ÿ๐‘˜ (๐บ) = ๐‘˜๐›พ(๐บ).
Theorem 179 (Dehgardi et al. [128]). For any graph ๐บ, if
๐›พ๐‘Ÿ๐‘˜ (๐บ) = ๐‘˜๐›ฝ(๐บ), then
(a) ๐‘๐‘Ÿ๐‘˜ (๐บ) โฉพ ๐›ฟ(๐บ);
(b) ๐‘๐‘Ÿ๐‘˜ (๐บ) โฉพ ๐›ฟ(๐บ) + 1 if ๐บ is vertex ๐‘˜-rainbow dominationcritical.
As far as we know, there are no more results on the ๐‘˜rainbow bondage number of graphs in the literature.
28
International Journal of Combinatorics
10. Results on Digraphs
Corollary 181. For a digraph ๐บ, ๐‘(๐บ) โฉฝ ๐›ฟโˆ’ (๐บ) + ฮ”(๐บ).
Although domination has been extensively studied in undirected graphs, it is natural to think of a dominating set
as a one-way relationship between vertices of the graph.
Indeed, among the earliest literatures on this subject, von
Neumann and Morgenstern [129] used what is now called
domination in digraphs to find solution (or kernels, which
are independent dominating sets) for cooperative ๐‘›-person
games. Most likely, the first formulation of domination by
Berge [130], in 1958, was given in the context of digraphs and,
only some years latter by Ore [131], in 1962, for undirected
graphs. Despite this history, examination of domination and
its variants in digraphs has been essentially overlooked (see
[4] for an overview of the domination literature). Thus, there
are few, if any, such results on domination for digraphs in the
literature.
The bondage number and its related topics for undirected
graph have become one of major areas both in theoretical
and applied researches. However, until recently, Carlson
and Develin [42], Shan and Kang [132], and Huang and
Xu [133, 134] studied the bondage number for digraphs,
independently. In this section, we introduce their results
for general digraphs. Results for some special digraphs such
as vertex-transitive digraphs are introduced in the next
section.
Since ๐›ฟโˆ’ (๐บ)
โฉฝ
(1/2)ฮ”(๐บ), from Corollary 181,
Conjecture 53 is valid for digraphs.
10.1. Upper Bounds for General Digraphs. Let ๐บ = (๐‘‰, ๐ธ)
be a digraph without loops and parallel edges. A digraph ๐บ
is called to be asymmetric if whenever (๐‘ฅ, ๐‘ฆ) โˆˆ ๐ธ(๐บ) then
(๐‘ฆ, ๐‘ฅ) โˆ‰ ๐ธ(๐บ) and to be symmetric if (๐‘ฅ, ๐‘ฆ) โˆˆ ๐ธ(๐บ) implies
(๐‘ฆ, ๐‘ฅ) โˆˆ ๐ธ(๐บ). For a vertex ๐‘ฅ of ๐‘‰(๐บ), the sets of outneighbors and in-neighbors of ๐‘ฅ are, respectively, defined as
๐‘๐บ+ (๐‘ฅ) = {๐‘ฆ โˆˆ ๐‘‰(๐บ) : (๐‘ฅ, ๐‘ฆ) โˆˆ ๐ธ(๐บ)} and ๐‘๐บโˆ’ (๐‘ฅ) = {๐‘ฅ โˆˆ
๐‘‰(๐บ) : (๐‘ฅ, ๐‘ฆ) โˆˆ ๐ธ(๐บ)}, the out-degree and the in-degree of
+
โˆ’
(๐‘ฅ) = |๐‘๐บ+ (๐‘ฅ)| and ๐‘‘๐บ
(๐‘ฅ) =
๐‘ฅ are, respectively, defined as ๐‘‘๐บ
|๐‘๐บ+ (๐‘ฅ)|. Denote the maximum and the minimum out-degree
(resp., in-degree) of ๐บ by ฮ”+ (๐บ) and ๐›ฟ+ (๐บ) (resp., ฮ”โˆ’ (๐บ) and
+
โˆ’
(๐‘ฅ)+๐‘‘๐บ
(๐‘ฅ), and
๐›ฟโˆ’ (๐บ)). The degree ๐‘‘๐บ(๐‘ฅ) of ๐‘ฅ is defined as ๐‘‘๐บ
the maximum and the minimum degrees of ๐บ are denoted by
ฮ”(๐บ) and ๐›ฟ(๐บ), respectively; that is, ฮ”(๐บ) = max{๐‘‘๐บ(๐‘ฅ) : ๐‘ฅ โˆˆ
๐‘‰(๐บ)} and ๐›ฟ(๐บ) = min{๐‘‘๐บ(๐‘ฅ) : ๐‘ฅ โˆˆ ๐‘‰(๐บ)}. Note that the
definitions here are different from the ones in the textbook
on digraphs; see for example, Xu [1].
A subset ๐‘† of ๐‘‰(๐บ) is called a dominating set if ๐‘‰(๐บ) = ๐‘†โˆช
๐‘๐บ+ (๐‘†), where ๐‘๐บ+ (๐‘†) is the set of out-neighbors of ๐‘†. Then, just
as for undirected graphs, ๐›พ(๐บ) is the minimum cardinality
of a dominating set, and the bondage number ๐‘(๐บ) is the
smallest cardinality of a set ๐ต of edges such that ๐›พ(๐บ โˆ’ ๐ต) >
๐›พ(๐บ) if such a subset ๐ต โŠ† ๐ธ(๐บ) exists. Otherwise, we put
๐‘(๐บ) = โˆž.
Some basic results for undirected graphs stated in
Section 3 can be generalized to digraphs. For example,
Theorem 18 is generalized by Carlson and Develin [42],
Huang and Xu [133], and Shan and Kang [132], independently,
as follows.
Theorem 180. Let ๐บ be a digraph and (๐‘ฅ, ๐‘ฆ) โˆˆ ๐ธ(๐บ). Then
โˆ’
(๐‘ฅ) โˆ’ |๐‘๐บโˆ’ (๐‘ฅ) โˆฉ ๐‘๐บโˆ’ (๐‘ฆ)|.
๐‘(๐บ) โฉฝ ๐‘‘๐บ(๐‘ฆ) + ๐‘‘๐บ
Corollary 182. For a digraph ๐บ, ๐‘(๐บ) โฉฝ (3/2)ฮ”(๐บ).
In the case of undirected graphs, the bondage number of
๐บ๐‘› = ๐พ๐‘› × ๐พ๐‘› achieves this bound. However, it was shown
by Carlson and Develin in [42] that if we take the symmetric
digraph ๐บ๐‘› , we have ฮ”(๐บ๐‘› ) = 4(๐‘›โˆ’1), ๐›พ(๐บ๐‘› ) = ๐‘›, and ๐‘(๐บ๐‘› ) โฉฝ
4(๐‘› โˆ’ 1) + 1 = ฮ”(๐บ๐‘› ) + 1. So this family of digraphs cannot
show that the bound in Corollary 182 is tight.
Corresponding to Conjecture 51, which is discredited
for undirected graphs and is valid for digraphs, the same
conjecture for digraphs can be proposed as follows.
Conjecture 183 (Carlson and Develin [42], 2006). ๐‘(๐บ) โฉฝ
ฮ”(๐บ) + 1 for any digraph ๐บ.
If Conjecture 183 is true, then the following results shows
that this upper bound is tight since ๐‘(๐พ๐‘› ×๐พ๐‘› ) = ฮ”(๐พ๐‘› ×๐พ๐‘› )+1
for a complete digraph ๐พ๐‘› .
In 2007, Shan and Kang [132] gave some tight upper
bounds on the bondage numbers for some asymmetric
digraphs. For example, ๐‘(๐‘‡) โฉฝ ฮ”(๐‘‡) for any asymmetric
directed tree ๐‘‡; ๐‘(๐บ) โฉฝ ฮ”(๐บ) for any asymmetric digraph ๐บ
with order at least 4 and ๐›พ(๐บ) โฉฝ 2. For planar digraphs, they
obtained the following results.
Theorem 184 (Shan and Kang [132], 2007). Let ๐บ be a
asymmetric planar digraph. Then ๐‘(๐บ) โฉฝ ฮ”(๐บ)+2; and ๐‘(๐บ) โฉฝ
โˆ’
(๐‘ฅ) โฉพ 3 for every vertex ๐‘ฅ with
ฮ”(๐บ) + 1 if ฮ”(๐บ) โฉพ 5 and ๐‘‘๐บ
๐‘‘๐บ(๐‘ฅ) โฉพ 4.
10.2. Results for Some Special Digraphs. The exact values and
bounds on ๐‘(๐บ) for some standard digraphs are determined.
Theorem 185 (Huang and Xu [133], 2006). For a directed cycle
๐ถ๐‘› and a directed path ๐‘ƒ๐‘› ,
3
๐‘ (๐ถ๐‘› ) = {
2
๐‘–๐‘“ ๐‘› ๐‘–๐‘  ๐‘œ๐‘‘๐‘‘,
๐‘–๐‘“ ๐‘› ๐‘–๐‘  ๐‘’V๐‘’๐‘›,
2 ๐‘–๐‘“ ๐‘› ๐‘–๐‘  ๐‘œ๐‘‘๐‘‘,
๐‘ (๐‘ƒ๐‘› ) = {
1 ๐‘–๐‘“ ๐‘› ๐‘–๐‘  ๐‘’V๐‘’๐‘›.
(76)
For the de Bruijn digraph ๐ต(๐‘‘, ๐‘›) and the Kautz digraph
๐พ(๐‘‘, ๐‘›),
๐‘ (๐ต (๐‘‘, ๐‘›)) = ๐‘‘ ๐‘–๐‘“ ๐‘› ๐‘–๐‘  ๐‘œ๐‘‘๐‘‘,
๐‘‘ โฉฝ ๐‘ (๐ต (๐‘‘, ๐‘›)) โฉฝ 2๐‘‘
๐‘–๐‘“ ๐‘› ๐‘–๐‘  ๐‘’V๐‘’๐‘›,
(77)
๐‘ (๐พ (๐‘‘, ๐‘›)) = ๐‘‘ + 1.
Like undirected graphs, we can define the total domination number and the total bondage number. On the total
bondage numbers for some special digraphs, the known
results are as follows.
International Journal of Combinatorics
29
Theorem 186 (Huang and Xu [134], 2007). For a directed
cycle ๐ถ๐‘› and a directed path ๐‘ƒ๐‘› , ๐‘๐‘‡ (๐‘ƒ๐‘› ) and ๐‘๐‘‡ (๐ถ๐‘› ) all do not
exist. For a complete digraph ๐พ๐‘› ,
๐‘๐‘‡ (๐พ๐‘› ) = โˆž ๐‘–๐‘“ ๐‘› = 1, 2,
๐‘๐‘‡ (๐พ๐‘› ) = 3
๐‘–๐‘“ ๐‘› = 3,
(78)
๐‘› โฉฝ ๐‘๐‘‡ (๐พ๐‘› ) โฉฝ 2๐‘› โˆ’ 3 ๐‘–๐‘“ ๐‘› โฉพ 4.
The extended de Bruijn digraph EB(๐‘‘, ๐‘›; ๐‘ž1 , . . . , ๐‘ž๐‘ ) and
the extended Kautz digraph EK(๐‘‘, ๐‘›; ๐‘ž1 , . . . , ๐‘ž๐‘ ) are introduced by Shibata and Gonda [135]. If ๐‘ = 1, then they
are the de Bruijn digraph B(๐‘‘, ๐‘›) and the Kautz digraph
K(๐‘‘, ๐‘›), respectively. Huang and Xu [134] determined their
total domination numbers. In particular, their total bondage
numbers for general cases are determined as follows.
Theorem 187 (Huang and Xu [134], 2007). If ๐‘‘ โฉพ 2 and ๐‘ž๐‘– โฉพ 2
for each ๐‘– = 1, 2, . . . , ๐‘, then
๐‘๐‘‡ (EB(๐‘‘, ๐‘›; ๐‘ž1 , . . . , ๐‘ž๐‘ )) = ๐‘‘๐‘ โˆ’ 1,
๐‘๐‘‡ (EK(๐‘‘, ๐‘›; ๐‘ž1 , . . . , ๐‘ž๐‘ )) = ๐‘‘๐‘ .
(79)
In particular, for the de Bruijn digraph B(๐‘‘, ๐‘›) and the Kautz
digraph K(๐‘‘, ๐‘›),
๐‘๐‘‡ (B (๐‘‘, ๐‘›)) = ๐‘‘ โˆ’ 1,
๐‘๐‘‡ (K (๐‘‘, ๐‘›)) = ๐‘‘.
(80)
Zhang et al. [136] determined the bondage number in
complete ๐‘ก-partite digraphs.
Theorem 188 (Zhang et al. [136], 2009). For a complete ๐‘กpartite digraph ๐พ๐‘›1 ,๐‘›2 ,...,๐‘›๐‘ก , where ๐‘›1 โฉฝ ๐‘›2 โฉฝ โ‹… โ‹… โ‹… โฉฝ ๐‘›๐‘ก ,
๐‘ (๐พ๐‘›1 ,๐‘›2 ,...,๐‘›๐‘ก )
๐‘š
{
{
{
{
{
{
{
= {4๐‘ก โˆ’ 3
{
{
๐‘กโˆ’1
{
{
{โˆ‘๐‘› + 2 (๐‘ก โˆ’ 1)
{
๐‘–
{๐‘–=1
๐‘–๐‘“ ๐‘›๐‘š = 1, ๐‘›๐‘š+1 โฉพ 2 ๐‘“๐‘œ๐‘Ÿ ๐‘ ๐‘œ๐‘š๐‘’,
๐‘š (1 โฉฝ ๐‘š < ๐‘ก)
๐‘–๐‘“ ๐‘›1 = ๐‘›2 = โ‹… โ‹… โ‹… = ๐‘›๐‘ก = 2,
๐‘œ๐‘กโ„Ž๐‘’๐‘Ÿ๐‘ค๐‘–๐‘ ๐‘’.
(81)
Since an undirected graph can be thought of as a symmetric digraph, any result for digraphs has an analogy for
undirected graphs in general. In view of this point, studying
the bondage number for digraphs is more significant than
for undirected graphs. Thus, we should further study the
bondage number of digraphs and try to generalize known
results on the bondage number and related variants for undirected graphs to digraphs, prove or disprove Conjecture 183.
In particular, determine the exact values of ๐‘(B(๐‘‘, ๐‘›)) for an
even ๐‘› and ๐‘๐‘‡ (๐พ๐‘› ) for ๐‘› โฉพ 4.
or |๐‘๐บโˆ’ [๐‘ฅ] โˆฉ ๐‘†| = 1 if ๐บ is a directed graph. The concept of an
efficient set was proposed by Bange et al. [137] in 1988. In the
literature, an efficient set is also called a perfect dominating
set (see Livingston and Stout [138] for an explanation).
From definition, if ๐‘† is an efficient dominating set of a
graph ๐บ, then ๐‘† is certainly an independent set and every
vertex not in ๐‘† is adjacent to exactly one vertex in ๐‘†.
It is also clear from definition that a dominating set ๐‘† is
efficient if and only if N๐บ[๐‘†] = {๐‘๐บ[๐‘ฅ] : ๐‘ฅ โˆˆ ๐‘†} for the
undirected graph ๐บ or N+๐บ[๐‘†] = {๐‘๐บ+ [๐‘ฅ] : ๐‘ฅ โˆˆ ๐‘†} for the
digraph ๐บ is a partition of ๐‘‰(๐บ), where the induced subgraph
by ๐‘๐บ[๐‘ฅ] or ๐‘๐บ+ [๐‘ฅ] is a star or an out-star with the root ๐‘ฅ.
The efficient domination has important applications in
many areas, such as error-correcting codes, and receives
much attention in the late years.
The concept of efficient dominating sets is a measure
of the efficiency of domination in graphs. Unfortunately, as
shown in [137], not every graph has an efficient dominating
set and, moreover, it is an NP-complete problem to determine
whether a given graph has an efficient dominating set. In
addition, it was shown by Clark [139] in 1993 that for a wide
range of ๐‘, almost every random undirected graph ๐บ โˆˆ
G(๐œ, ๐‘) has no efficient dominating sets. This means that
undirected graphs possessing an efficient dominating set are
rare. However, it is easy to show that every undirected graph
has an orientation with an efficient dominating set (see Bange
et al. [140]).
In 1993, Barkauskas and Host [141] showed that determining whether an arbitrary oriented graph has an efficient
dominating set is NP-complete. Even so, the existence of
efficient dominating sets for some special classes of graphs has
been examined; see, for example, Dejter and Serra [142] and
Lee [143] for Cayley graph, Gu et al. [144] for meshes and tori,
Obradovicฬ et al. [145], Huang and Xu [36], Reji Kumar and
MacGillivray [146] for circulant graphs, Harary graphs and
tori, and Van Wieren et al. [147] for cube-connected cycles.
In this section, we introduce results of the bondage
number for some graphs with efficient dominating sets, due
to Huang and Xu [36].
11.1. Results for General Graphs. In this subsection, we introduce some results on bondage numbers obtained by applying
efficient dominating sets. We first state the two following
lemmas.
Lemma 189. For any ๐‘˜-regular graph or digraph ๐บ of order ๐‘›,
๐›พ(๐บ) โฉพ ๐‘›/(๐‘˜ + 1), with equality if and only if ๐บ has an efficient
dominating set. In addition, if ๐บ has an efficient dominating
set, then every efficient dominating set is certainly a ๐›พ-set, and
vice versa.
11. Efficient Dominating Sets
Let ๐‘’ be an edge and ๐‘† a dominating set in ๐บ. We say that
๐‘’ supports ๐‘† if ๐‘’ โˆˆ (๐‘†, ๐‘†), where (๐‘†, ๐‘†) = {(๐‘ฅ, ๐‘ฆ) โˆˆ ๐ธ(๐บ) :
๐‘ฅ โˆˆ ๐‘†, ๐‘ฆ โˆˆ ๐‘†}. Denote by ๐‘ (๐บ) the minimum number of edges
which support all ๐›พ-sets in ๐บ.
A dominating set ๐‘† of a graph ๐บ is called to be efficient if for
every vertex ๐‘ฅ in ๐บ, |๐‘๐บ[๐‘ฅ] โˆฉ ๐‘†| = 1 if ๐บ is a undirected graph
Lemma 190. For any graph or digraph ๐บ, ๐‘(๐บ) โฉพ ๐‘ (๐บ), with
equality if ๐บ is regular and has an efficient dominating set.
30
International Journal of Combinatorics
A graph ๐บ is called to be vertex-transitive if its automorphism group Aut(๐บ) acts transitively on its vertex-set ๐‘‰(๐บ).
A vertex-transitive graph is regular. Applying Lemmas 189
and 190, Huang and Xu obtained some results on bondage
numbers for vertex-transitive graphs or digraphs.
Theorem 191 (Huang and Xu [36], 2008). For any vertextransitive graph or digraph ๐บ of order ๐‘›,
๐‘›
{
โŒ‰ ๐‘–๐‘“ ๐บ ๐‘–๐‘  ๐‘ข๐‘›๐‘‘๐‘–๐‘Ÿ๐‘’๐‘๐‘ก๐‘’๐‘‘,
โŒˆ
{
{
{ 2๐›พ (๐บ)
๐‘ (๐บ) โฉพ {
{
๐‘›
{
{โŒˆ
โŒ‰ ๐‘–๐‘“ ๐บ ๐‘–๐‘  ๐‘‘๐‘–๐‘Ÿ๐‘’๐‘๐‘ก๐‘’๐‘‘.
{ ๐›พ (๐บ)
(82)
Theorem 192 (Huang and Xu [36], 2008). Let ๐บ be a ๐‘˜-regular
graph of order ๐‘›. Then
๐‘(๐บ) โฉฝ ๐‘˜ if ๐บ is undirected and ๐‘› โฉพ ๐›พ(๐บ)(๐‘˜ + 1) โˆ’ ๐‘˜ + 1;
๐‘(๐บ) โฉฝ ๐‘˜ + 1 + โ„“ if ๐บ is directed and ๐‘› โฉพ ๐›พ(๐บ)(๐‘˜ + 1) โˆ’ โ„“
with 0 โฉฝ โ„“ โฉฝ ๐‘˜ โˆ’ 1.
Next, we introduce a better upper bound on ๐‘(๐บ). To this
aim, we need the following concept, which is a generalization
of the concept of the edge-covering of a graph ๐บ. For ๐‘‰๓ธ€  โŠ†
๐‘‰(๐บ) and ๐ธ๓ธ€  โŠ† ๐ธ(๐บ), we say that ๐ธ๓ธ€  covers ๐‘‰๓ธ€  and call ๐ธ๓ธ€ 
an edge-covering for ๐‘‰๓ธ€  if there exists an edge (๐‘ฅ, ๐‘ฆ) โˆˆ ๐ธ๓ธ€  for
any vertex ๐‘ฅ โˆˆ ๐‘‰๓ธ€  . For ๐‘ฆ โˆˆ ๐‘‰(๐บ), let ๐›ฝ๓ธ€  [๐‘ฆ] be the minimum
cardinality over all edge-coverings for ๐‘๐บโˆ’ [๐‘ฆ].
Theorem 193 (Huang and Xu [36], 2008). If ๐บ is a ๐‘˜-regular
graph with order ๐›พ(๐บ)(๐‘˜ + 1), then ๐‘(๐บ) โฉฝ ๐›ฝ๓ธ€  [๐‘ฆ] for any ๐‘ฆ โˆˆ
๐‘‰(๐บ).
The upper bound on ๐‘(๐บ) given in Theorem 193 is tight in
view of ๐‘(๐ถ๐‘› ) for a cycle or a directed cycle ๐ถ๐‘› (see Theorems
1 and 185, resp.).
It is easy to see that for a ๐‘˜-regular graph ๐บ, โŒˆ(๐‘˜ + 1)/2โŒ‰ โฉฝ
๐›ฝ๓ธ€  [๐‘ฆ] โฉฝ ๐‘˜ when ๐บ is undirected and ๐›ฝ๓ธ€  [๐‘ฆ] = ๐‘˜ + 1 when ๐บ is
directed. By this fact and Lemma 189, the following theorem
is merely a simple combination of Theorems 191 and 193.
Theorem 194 (Huang and Xu [36], 2008). Let ๐บ be a vertextransitive graph of degree ๐‘˜. If ๐บ has an efficient dominating
set, then
โŒˆ
๐‘˜+1
โŒ‰ โฉฝ ๐‘ (๐บ) โฉฝ ๐‘˜
2
๐‘–๐‘“ ๐บ ๐‘–๐‘  ๐‘ข๐‘›๐‘‘๐‘–๐‘Ÿ๐‘’๐‘๐‘ก๐‘’๐‘‘,
(83)
๐‘ (๐บ) = ๐‘˜ + 1 ๐‘–๐‘“ ๐บ ๐‘–๐‘  ๐‘‘๐‘–๐‘Ÿ๐‘’๐‘๐‘ก๐‘’๐‘‘.
Theorem 195 (Huang and Xu [36], 2008). If ๐บ is an undirected vertex-transitive cubic graph with order 4๐›พ(๐บ) and girth
๐‘”(๐บ) โฉฝ 5, then ๐‘(๐บ) = 2.
The proof of Theorem 195 leads to a byproduct. If ๐‘” =
5, let (๐‘ข1 , ๐‘ข2 , ๐‘ข3 , ๐‘ข4 , ๐‘ข5 ) be a cycle, and let D๐‘– be the family
of efficient dominating sets containing ๐‘ข๐‘– for each ๐‘– โˆˆ
{1, 2, 3, 4, 5}. Then D๐‘– โˆฉ D๐‘— = 0 for ๐‘– = 1, 2, . . . , 5. Then ๐บ has
at least 5๐‘  efficient dominating sets, where ๐‘  = ๐‘ (๐บ). But there
are only 4s (=ns/๐›พ) distinct efficient dominating sets in ๐บ. This
contradiction implies that an undirected vertex-transitive
cubic graph with girth five has no efficient dominating sets.
But a similar argument for ๐‘”(๐บ) = 3, 4 or ๐‘”(๐บ) โฉพ 6 could
not give any contradiction. This is consistent with the result
that CCC(๐‘›), a vertex-transitive cubic graph with girth ๐‘› if
3 โฉฝ ๐‘› โฉฝ 8, or girth 8 if ๐‘› โฉพ 9, has efficient dominating sets
for all ๐‘› โฉพ 3 except ๐‘› = 5 (see Theorem 199 in the following
subsection).
11.2. Results for Cayley Graphs. In this subsection, we use
Theorem 194 to determine the exact values or approximative
values of bondage numbers for some special vertex-transitive
graphs by characterizing the existence of efficient dominating
sets in these graphs.
Let ฮ“ be a nontrivial finite group, ๐‘† a nonempty subset of
ฮ“ without the identity element of ฮ“. A digraph ๐บ is defined as
follows:
๐‘‰ (๐บ) = ฮ“,
(๐‘ฅ, ๐‘ฆ) โˆˆ ๐ธ (๐บ) โ‡โ‡’ ๐‘ฅโˆ’1 ๐‘ฆ โˆˆ ๐‘†
for any ๐‘ฅ, ๐‘ฆ โˆˆ ฮ“
(84)
is called a Cayley digraph of the group ฮ“ with respect to
๐‘†, denoted by ๐ถฮ“ (๐‘†). If ๐‘†โˆ’1 = {๐‘ โˆ’1 : ๐‘  โˆˆ ๐‘†} = ๐‘†; then
๐ถฮ“ (๐‘†) is symmetric and is called a Cayley undirected graph,
a Cayley graph for short. Cayley graphs or digraphs are
certainly vertex-transitive (see Xu [1]).
A circulant graph ๐บ(๐‘›; ๐‘†) of order ๐‘› is a Cayley graph
๐ถ๐‘๐‘› (๐‘†), where ๐‘๐‘› = {0, . . . , ๐‘› โˆ’ 1} is the addition group of
order ๐‘› and ๐‘† is a nonempty subset of ๐‘๐‘› without the identity
element and, hence, is a vertex-transitive digraph of degree
|๐‘†|. If ๐‘†โˆ’1 = ๐‘†, then ๐บ(๐‘›; ๐‘†) is an undirected graph. If ๐‘† = {1, ๐‘˜},
where 2 โฉฝ ๐‘˜ โฉฝ ๐‘› โˆ’ 2, we write ๐บ(๐‘›; 1, ๐‘˜) for ๐บ(๐‘›; {1, ๐‘˜}) or
๐บ(๐‘›; {±1, ±๐‘˜}) and call it a double-loop circulant graph.
Huang and Xu [36] showed that, for directed ๐บ =
๐บ(๐‘›; 1, ๐‘˜), โŒˆ๐‘›/3โŒ‰ โฉฝ ๐›พ(๐บ) โฉฝ โŒˆ๐‘›/2โŒ‰ and ๐บ has an efficient
dominating set if and only if 3 | ๐‘› and ๐‘˜ โ‰ก 2 (mod 3); for
directed ๐บ = ๐บ(๐‘›; 1, ๐‘˜) and ๐‘˜ =ฬธ ๐‘›/2, โŒˆ๐‘›/5โŒ‰ โฉฝ ๐›พ(๐บ) โฉฝ โŒˆ๐‘›/3โŒ‰
and ๐บ has an efficient dominating set if and only if 5 | ๐‘› and
๐‘˜ โ‰ก ±2 (mod 5). By Theorem 194, we obtain the bondage
number of a double loop circulant graph if it has an efficient
dominating set.
Theorem 196 (Huang and Xu [36], 2008). Let ๐บ be a doubleloop circulant graph ๐บ(๐‘›; 1, ๐‘˜). If ๐บ is directed with 3 | ๐‘›
and ๐‘˜ โ‰ก 2 (mod 3), or ๐บ is undirected with 5 | ๐‘› and
๐‘˜ โ‰ก ±2 (mod 5), then ๐‘(๐บ) = 3.
The ๐‘š × ๐‘› torus is the Cartesian product ๐ถ๐‘š × ๐ถ๐‘› of
two cycles and is a Cayley graph ๐ถ๐‘๐‘š ×๐‘๐‘› (๐‘†), where ๐‘† =
{(0, 1), (1, 0)} for directed cycles and ๐‘† = {(0, ±1), (±1, 0)} for
undirected cycles and, hence, is vertex-transitive. Gu et al.
[144] showed that the undirected torus ๐ถ๐‘š ×๐ถ๐‘› has an efficient
dominating set if and only if both ๐‘š and ๐‘› are multiples
of 5. Huang and Xu [36] showed that the directed torus
๐ถ๐‘š × ๐ถ๐‘› has an efficient dominating set if and only if both
๐‘š and ๐‘› are multiples of 3. Moreover, they found a necessary
condition for a dominating set containing the vertex (0, 0)
in ๐ถ๐‘š × ๐ถ๐‘› to be efficient and obtained the following
result.
International Journal of Combinatorics
Theorem 197 (Huang and Xu [36], 2008). Let ๐บ = ๐ถ๐‘š × ๐ถ๐‘› .
If ๐บ is undirected and both ๐‘š and ๐‘› are multiples of 5 or if ๐บ is
directed and both ๐‘š and ๐‘› are multiples of 3, then ๐‘(๐บ) = 3.
The hypercube ๐‘„๐‘› is the Cayley graph ๐ถฮ“ (๐‘†),
where ฮ“ = ๐‘2 × โ‹… โ‹… โ‹… × ๐‘2 = (๐‘2 )๐‘› and ๐‘† =
{100 โ‹… โ‹… โ‹… 0, 010 โ‹… โ‹… โ‹… 0, . . . , 00 โ‹… โ‹… โ‹… 01}. Lee [143] showed that
๐‘„๐‘› has an efficient dominating set if and only if ๐‘› = 2๐‘š โˆ’ 1
for a positive integer ๐‘š. By Theorem 194, the following result
is obtained immediately.
Theorem 198 (Huang and Xu [36], 2008). If ๐‘› = 2๐‘š โˆ’ 1 for a
positive integer ๐‘š, then 2๐‘šโˆ’1 โฉฝ ๐‘(๐‘„๐‘› ) โฉฝ 2๐‘š โˆ’ 1.
Problem 10. Determine the exact value of ๐‘(๐‘„๐‘› ) for ๐‘› = 2๐‘š โˆ’1.
The ๐‘›-dimensional cube-connected cycle, denoted by
CCC(๐‘›), is constructed from the ๐‘›-dimensional hypercube
๐‘„๐‘› by replacing each vertex ๐‘ฅ in ๐‘„๐‘› with an undirected cycle
๐ถ๐‘› of length ๐‘› and linking the ๐‘–th vertex of the ๐ถ๐‘› to the ๐‘–th
neighbor of ๐‘ฅ. It has been proved that CCC(๐‘›) is a Cayley
graph and, hence, is a vertex-transitive graph with degree 3.
Van Wieren et al. [147] proved that CCC(๐‘›) has an efficient
dominating set if and only if ๐‘› =ฬธ 5. From Theorems 194 and
195, the following result can be derived.
Theorem 199 (Huang and Xu [36], 2008). Let ๐บ = ๐ถ๐ถ๐ถ(๐‘›)
be the ๐‘›-dimensional cube-connected cycles with ๐‘› โฉพ 3 and
๐‘› =ฬธ 5. Then ๐›พ(๐บ) = ๐‘›2๐‘›โˆ’2 and 2 โฉฝ ๐‘(๐บ) โฉฝ 3. In addition,
๐‘(๐ถ๐ถ๐ถ(3)) = ๐‘(๐ถ๐ถ๐ถ(4)) = 2.
Problem 11. Determine the exact value of ๐‘(CCC(๐‘›)) for ๐‘› โฉพ
5.
Acknowledgments
This work was supported by NNSF of China (Grants nos.
11071233, 61272008). The author would like to express his
gratitude to the anonymous referees for their kind suggestions and comments on the original paper, to Professor
Sheikholeslami and Professor Jafari Rad for sending the
references [128] and [81], which resulted in this version.
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Journal of
Discrete Mathematics
Journal of
Volume 2014
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Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
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Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
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Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014