Download Some Domination Parameters of Prime Square Dominating

DOI 10.4010/2016.1925
ISSN 2321 3361 © 2016 IJESC
Research Article
Volume 6 Issue No. 7
Some Domination Parameters of Prime Square Dominating Graphs
A. Sudhakaraiah *, V. Raghava Lashmi1, B. Narayana2
Department of Mathematics, S. V. University, Tirupati, Andhra Pradesh, India*
Department of S&H (Mathematics), Vijaya Institute of Technology for Women, Vijayawada, A. P., India1
Department of Basic Science and Humanities, St, Mary’s Women’s Engineering College, Guntur, A. P. India2
Abstract
The purpose of this paper is to typify Prime Square Dominating Graphs and to discuss some of their domination parameters. Theorems
are proposed to establish domination number and complementary tree domination number of some of the Prime Square Dominating
Graphs. Moreover, conditions sufficient for Prime Square Dominating Graph characterization of a certain class of graphs are put forth.
Keywords: Complementary Tree Dominating Set, Complementary Tree Domination Number, Dominating Set, Prime Square
Dominating Graph.
I. INTRODUCTION
Graph theory is one of the most beautiful branches of
modern mathematics with a wide range of research scope. Many
real life situations can be modeled as a graph theory problem and
can be represented diagrammatically with a set of points called
as vertices joined together with lines called as edges. For
example, in network problems, network places can be
represented by vertices and network links can be represented by
edges, so that real life situations can be understood in a better
demeanor.
Some of the important areas of research in graph theory
are domination and the graph labeling. The domination problem
in a graph is to find a minimum sized vertex set D such that
every vertex not in D is adjacent to at least one vertex in D. The
domination number of the graph G is denoted by γ (G) and is the
minimum cardinality of a dominating set of G. Dominating sets
play an imperative role in algorithms and in combinatorics.
Complementary tree dominating set problem is the
problem of finding complementary tree dominating set of
specific size for the given graph. A dominating set
S
 V of a graph G with vertex set V (G) and edge set E (G) is a
complementary tree dominating set if the induced sub graph
˂ V – S > is a tree. Complementary tree domination number is
the minimum cardinality of a complementary tree dominating set
of G. It is denoted by
(G). The notion of complementary
tree dominating set is due to
S. Muttamai et al. [1]. Some
results pertaining to the bounds of Complementary tree
domination number are obtained by them. In [2] and [3]
Complementary tree domination number of some specified class
of interval graphs and circular arc graphs are obtained.
Labeling of a graph plays a vital role in graph theory.
Seod M.A.and Youssef M.Z [4] and Joseph A. Gallian [5] deal
with graph labeling besides many others. A labeling of a graph
G is an assignment of distinct positive integers to its vertices. A
graph is a prime square dominating graph, if the vertices of
graph G are labeled with positive integers such that the vertex
labeled with composite number c is adjacent to the vertex named
with prime number p if and only if p2/c (p2 divides c). In this
paper, the study of complementary tree domination number of
International Journal of Engineering Science and Computing, July 2016
graphs is extended to a special class of graphs namely, prime
square dominating graphs.
Unless otherwise specified, throughout the paper the
graph G(V, E) is a finite and connected graph with the vertex set
V(G) and edge set E(G) and standard definitions of graph theory
as found in [6] are followed.
II. IMPORTANT RESUITS
Result 2.1: For a connected graph G of order
(G) ≤ k-1.
k ≥ 2,
Result 2.2: For a connected graph G of order
(G) ≤
(G).
k ≥ 2, γ
Result 2.3: For a connected graph, every pendant vertex is a
member of all ctd – sets [1].
Result 2.4: Every prime square dominating graph is a finite and
simple graph.
Result 2.5: Every prime square dominating graph is a prime
dominating graph. The converse is not true.
Result 2.5: Every graph is a bipartite graph if and only if the
graph is a prime square dominating graph [7].
III. DISCUSSIONS
CHARECTERIZATION OF GRAPHS AS PRIME
SQUARE DOMINATING GRAPHS
3. 1 THEOREM: Every cycle graph with even number of
vertices is a prime square dominating graph.
Proof: Let
be a cycle graph with vertex set V= { , ,
,...,
}. Label the vertices , ,...,
with the prime
numbers
,
,
,...,
respectively and the vertices
, ,
,...,
with composite numbers , , ,...,
respectively, so that
divides
,
divides
,
divides
……….. and
divides
&
.
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From the labeling of the graph, it is clear that the
vertices of graph
are labeled with positive integers such that
the vertex labeled with composite number c is adjacent to the
vertex labeled with prime number p if p2/c. By the definition of
prime square dominating graph, it follows that the graph
is a
prime square dominating graph. Therefore, the vertices of every
cycle graph with even number of vertices can be labeled in such
a way that the graph becomes a prime square dominating graph.
Hence the result follows.
2
225
2
_
1
5
Note: A cycle graph with odd number of vertices is a not a
prime square dominating graph [7].
3
Experimental problem: Consider a cycle graph
which will be as follows:
Prime square dominating graph,
Cycle graph
Label the odd vertices ,
of the graph with
prime numbers 2, 3, 5 and the even vertices , ,
with composite numbers 36, 225 & 100 respectively.
Clearly, vertex labeled with prime number 2 is
adjacent to vertices labeled with composite numbers
100 & 36 and moreover 22 divides36 & 100.
Similarly, 3 is adjacent to vertices labeled with
composite numbers 225 & 36 and moreover 32
divides 225 & 36 and 5 is adjacent to vertices labeled
with composite numbers 225 & 100 and moreover 52
divides 225 &100. Implies, G is a prime square
dominating graph.
Therefore, labeling of the graph
is done in
such a way that the graph G is a prime square
dominating graph.
International Journal of Engineering Science and Computing, July 2016
3.2 THEOREM: Every tree is a prime square
dominating graph.
Proof:
Let T be a tree of height ‘h’. Here two cases will
arise.
Case i: h may be
even. Partition the vertex set into two subsets
Such that
= set of all vertices
at level o, 2, 4, ……, h and
= set of all vertices at level 1, 3, 5, ……, h-1
Case ii: h may be odd. Partition the vertex set V into
two subsets
Such that
=
set of all vertices at level o, 2, 4, ……, h-1 and
= set of all vertices at level 1, 3, 5, ……, h.
In both the cases, the vertex set V of the graph
is partitioned into two sets
and
so that every
edge of the graph has one end vertex in and another
end vertex in
Label all the vertices in
ith
prime numbers and all the vertices in
with
composite numbers depending upon the labeling of
the vertices in
such that the vertex labeled with
composite number ‘c’ is adjacent to the vertex labeled
with prime number ‘p’ if and only if p2/c. By the
definition of prime square dominating graph, the tree
T becomes a prime square dominating graph. It
follows that, every tree is a prime square dominating
graph.
Experimental problem: Consider a tree as follows:
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Tree with height 3
Partition the vertex set into two subsets
that
= set of all vertices at level o & 2 and
7
of all vertices at level 1 & 3. Implies
{
and
{
Such
= set
=
Prime Square Dominating Graph
=
Label the vertices
of the tree ‘T’
with prime numbers 11, 3, 5, 2 & 7 respectively and
the vertices
with composite numbers 27225, 23716, 18, 27, 8, 12,
49 & 98 respectively. Clearly vertex labeled with
composite number 27225 is adjacent to vertices
labeled with prime numbers 11, 3 & 5 and moreover
112, 32 & 52 divide 27225, vertex labeled with
composite number 23716 is adjacent to vertices
labeled with prime numbers 11, 2 & 7 and moreover
112, 22 & 72 divide 23716. Similarly, 18 & 27 are
adjacent to the vertex labeled with prime number 3
and 32 divides 18 & 27; 8 & 12 are adjacent to the
vertex labeled with prime number 2 and 22 divides 8
& 12 and 49 & 98 are adjacent to the vertex labeled
with prime number 7 and 72 divides 49 & 98. From
the definition of the prime square dominating graph, it
follows that T is a prime square dominating graph
Therefore, labeling of every Tree
can done in
such a way that the tree becomes a prime square
dominating graph. In this case, the corresponding
prime square dominating graph will be as follows
International Journal of Engineering Science and Computing, July 2016
Domination Number and Complementary Tree
Domination number 0f Prime Square Dominating
Graphs
3. 3 THEOREM: Let G be a prime square
dominating
graph
with
vertex
set
V= { , , ,…, }, n ≥ 3. Let the vertices with
even suffixes be labeled with prime numbers ,
….respectively starting with and the vertices
with odd suffixes be labeled with composite
numbers ,
…………… respectively starting
with such that
divides only
&
for 1≤ i
≤ n and for even values of i.. Then γ (G) = n/2 if n is
even and
if n is odd and
(G) = n-2
Proof: Let the vertex set V={ , , ,…, } of
the graph G be labeled with composite numbers and
prime numbers satisfying the conditions mentioned in
the theorem. It is supposed that
divides only
&
for
1≤ i ≤ n and for even values
of i. It follows that
divides only &
divides only &
divides only &
……………………
divides only
when n is even and
divides only
& when n is odd
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As the graph is a prime square dominating graph,
the edges of the graph are { , },
{ , }, {
,
, { , }, ………………..., {
,
,{
,
}, {
,
if n is even and { , }, {
, }, {
,
, {
,
}, ………….., {
,
,{
,
},
{
,
,
{
, } if n is odd. In both the cases all the edges
are distinct and they form a path graph. Hence, γ (G)
= n/2 if n is even and
if n is odd and
(G) =
n-2 [1].
Experimental problem: Consider the prime square
dominating graph G with vertex set
V=
{ 9, 3, 36, 2, 100, 5, 1225, 7}. It will be as follows:
not adjacent to any other vertex, except
for
k = 2, 3, 4, ……, n. Implies d( ) = 1 for k = 2, 3, 4,
……, n. The vertices , ,... …..,
are pendant
vertices. Every ctd – 18
set of a graph consists of every
pendant23716
vertex of the graph. Every ctd – set of graph
G consists of
, ,... ………, . Implies,
(G) ≥ n – 1. But for any graph of order n≥2,
(G) ≤ n – 1. It follows that,
(G) = n – 1. The
minimum
, ,... …..,
. As vertex
12 ctd – set is
is adjacent to
k = 2, 3, 4,……, n,
inates
27
all the remaining vertices of the graph. Therefore, the
set { 98} 27225
is the dominating set of the graph and it is a
minimal dominating set of the graph. Hence, γ (G) =
1
8
9
3
2
5
Experimental problem: Consider the prime square
dominating graph G with vertex set
V = { 2, 8,
16, 40, 60, }. It will be as follows
36
6
100
1225
8
7
16
Prime Square Dominating Graph
Here the vertex labeling satisfies all the conditions
mentioned in the theorem, where
,
,
. Also, the order of the graph, n = 8( even ).
Hence, γ (G) = 8/2 = 4 and
(G) = 9 - 2 = 7.
3. 4 THEOREM: Let the vertex set
V
= { , ,
,...,
}, n ≥ 2 of a prime square
dominating graph G be such that a vertex , 1≤ i ≤ n
is labeled with a prime number
and the remaining
all vertices are labeled with composite numbers
,
,... , such that
divides all the vertices
labeled with composite numbers. Then G is a prime
square dominating graph with γ(G)=1 and
(G)=n1.
Proof: Let the vertex set V={ , , ,…, } of
the prime square dominating graph G be labeled with
composite numbers and prime numbers satisfying the
conditions mentioned in the theorem. Implies, vertex
is adjacent to
k = 2, 3, 4, ……, n. As there is
no other vertex labeled with prime, except
is
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2
40
60
Prime Square Dominating Graph
The graph satisfies all the conditions stated in the
above theorem for
=2,
,
,
,
and moreover order of the graph, n = 5.
Clearly, γ (G) =1 and
(G) = 4
where
Minimal
dominating
set
=
{
2
}
& minimal ctd – set is {8, 16, 40, 60 }
3. 5 THEOREM: Let the vertex set
V
= { , , ,..., }, n ≥ 4 of a graph be such that
the vertices
&
1≤ i, j ≤ n are labeled with a
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prime
number
and composite number
and some of the remaining vertices
are labeled with composite numbers and some with
prime
numbers
so
that
and all the
squares of primes divide only
. Then
γ (G) = 2 and
(G) = n-2.
vertices labeled with 3, 5, 7, 44, 40, 16, 8 are pendant
vertices and the vertices
= 2 & = 44100 satisfy
all the conditions mentioned in the theorem. Hence
γ (G) = 2 &
(G) = n – 2= 9 - 2 = 7, Where
Minimal dominating set = {2, 44100} & minimum
ctd – set is {3, 5, 7, 44, 40, 16, 8 }.
Prime square
dominating graph in this case will be as follows
Proof:
Let
the
vertex
set
V = { , , ,………, }, of a prime square
dominating graph G be labeled with composite
numbers and prime numbers satisfying the conditions
mentioned in the theorem.
By hypothesis, except , the squares of all
the other primes divide only one composite number
. So, all the vertices labeled with primes except
are adjacent only to and are pendant vertices. Let
them be p in number. Moreover, all the composite
numbers, except
are divisible only by
. All
vertices labeled with composite numbers other than
are pendant vertices. Let them be q in number. is
adjacent to all the vertices labeled with composite
numbers and is adjacent to all the vertices labeled
with prime numbers. Therefore,
and are not
pendant vertices. The order of the graph is n. Implies
p + q + 2 = n and
p + q = n-2. G has n - 2
pendant vertices. Every ctd – set consists of every
pendant vertex of the graph. Every ctd – set of graph
G consists of
n - 2 vertices. It follows
that
(G) ≥ n – 2
Let S = V - {
}. As vertex is adjacent
to every vertex labeled with composite number and
.is adjacent to every vertex labeled with prime
integer, S is a dominating set. Moreover,
is
adjacent to
. It follows that the induced subgraph ˂
V – S ˃ is a tree. S is a complementary tree
dominating set. Implies
(G) ≤ n – 2.
Therefore,
(G) = n – 2 and the minimal
complementary tree dominating set is V - {
}. It
is obvious that the set
{
} is the minimal
dominating set and Therefore, γ (G) = 2.
Experimental problem: Consider the prime square
dominating graph with vertex set
V = {2, 3,
5, 7, 44, 44100, 40, 16, 8}. Here 22 divides 44, 44100,
40, 16 & 8; 32, 52 & 72 divide only 44100. The
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44
2
44100
3
40
5
16
7
8
Prime Square Dominating Graph
IV. Conclusion
Resolving the domination parameters of some special
classes of prime square dominating graphs has been
the main focus of the paper. Some categorized graphs
have been chosen in the process of examination. In
future, efforts will be put to find dominator chromatic
number
and other domination parameters of prime square
dominating graphs.
V. REFERENCES
[1] S. Muthammai, M. Bhanumathi and P. Vidhya,
Complimentary tree domination number of a graph,
International MathematicalForum,Vol.6, no. 26 (2011), 1273
– 1282.
[2] Dr. A. Sudhakaraiah, V. Raghava Lakshmi, V.Rama Latha
and T.Venkateswarlu,
Complimentary tree domination
number of interval graphs, International Journal of
Advanced Research in Computer Science and Software
Engineering, Vol. 3, No. 6 (2013), 1338-1342.
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[3]
Dr. A. Sudhakaraiah, V. Raghava Lakshmi, and
T.Venkateswarlu, Complimentary tree domination number
of circular- arc graphs, International Journal of Engineering
and Innovative Technology (IJEIT), Volume 3, Issue 2,
August 2013, pg. 229-232.
[4] Seod M.A.and Youssef M.Z., “On prime labeling of graphs”,
Congr. Number.141(1999), 203-215.
[5] Joseph A. Gallian, “A Dynamic Survey of graph labeling”,
the electronic journal of combinatorics #D S 6, 2005.
[6] F.Harrary, Graph Theory, Addition - Wesley Reading Mass,
1969.
[7] A. Sudhakaraiah and B. Narayana, A note on prime square
dominating graph, , IOSR Journal of Business and
Management (IOSRJBM) Volume 2, Issue 6 (Sep,-Oct.
2012), PP 10-13.
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