Download The 3*-connected property of the pyramid networks

The 3*-connected property
of the pyramid networks
Yuan-Hsiang Teng, Tzu-Liang Kung, Lih-Hsing Hsu
Computers and Mathematics with Applications 60 (2010)
2360–2363
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Abstract

We prove that the pyramid network PM[n] is
3*-connected and super connected for n ≥ 1.
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Definition and Notation for a Graph
G=(V,E) is a graph if V is a finite set and E is
a subset of {(a,b) | (a,b) is an unordered pair of
V}.
 V is the node set and E is the edge set of G.

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Hamiltonian Properties
A hamiltonian path is a path such that its
nodes are distinct and span V.
 A hamiltonian cycle is a cycle such that its
nodes are distinct except for the first node and
the last node and span V.
 A hamiltonian graph is a graph with a
hamiltonian cycle.

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Connectivity

The connectivity of G, κ(G) is the minimum
number of vertices whose removal leaves the
remaining graph disconnected or trivial.
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Container
A k-container C(u,v) in a graph G is a set of k
disjoint paths joining u to v.
 A k*-container C(u,v) in a graph G is a kcontainer such that every vertex of G is on
some path in C(u,v).

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k*-connected





A graph G is k*-connected if there exists a k*container between any two distinct vertices in G.
A graph G is 1*-connected if and only if it is
hamiltonian connected.
A graph G is 2*-connected if it is hamiltonian.
The study of k*-connected graph is motivated by the
3*-connected graphs proposed by Albert et al.
M. Albert, E.R.L. Aldred, D. Holton, J. Sheehan, On 3*-connected graphs, Australasian
Journal of Combinatorics 24 (2001) 193-207.
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Super connected

A graph G is super connected if it is i*connected for all 1≤ i ≤ κ(G).
• H.C. Hsu, C.K. Lin, H.M. Huang, L.H. Hsu, The spanning connectivity of the (n, k)star graphs, International Journal of Foundations of Computer Science
17 (2006) 415–434.
• C.K. Lin, H.M. Huang, L.H. Hsu, The super connectivity of the pancake graphs and
the super laceability of the star graphs, Theoretical Computer Science
339 (2005) 257–271.
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Introduction





κ(PM[n])=3
Sarbazi-Azad et al. andWu et al. studied the
hamiltonicity and the hamiltonian connectivity of the
pyramid networks.
Thus the pyramid network is 1*-connected and 2*connected.
H. Sarbazi-Azad, H.M. Ould-Khaoua, L.M. Mackenzie, Algorithmic construction
of hamiltonians in pyramids, Information Processing Letters 80 (2001) 75-79.
R.Y. Wu, D.R. Duh, Hamiltonicity of the pyramid network with or without fault,
Journal fo Information Science and Engineering 25 (2009) 531-542.
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Introduction



In this paper, we study the 3*-connected
property of the pyramid network.
We prove that any pyramid network is 3*connected.
Moreover, the pyramid network is super
connected.
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The pyramid networks



An n-dimensional pyramid network PM[n] is
a hierarchy structure based on mesh networks.
The subgraph induced by all vertices in the ith layer of a PM[n] is a mesh network
M(2i, 2i).
V (PM[n]) = {(k; x; y) | 0 ≤k ≤ n; 1 ≤ x ≤2k;
1 ≤y ≤2k}.
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The pyramid networks

Two vertices (k1; x1; y1) and (k2; x2; y2) in
PM[n] are adjacent if they satisfy one of the
following conditions:
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The pyramid networks PM[2]
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Lemma 1

A pyramid network PM[n] is hamiltonian for
n ≥1.
• H. Sarbazi-Azad, H.M. Ould-Khaoua, L.M. Mackenzie, Algorithmic construction
of Hamiltonians in pyramids, Information Processing Letters 80 (2001)
75–79.
• R.Y. Wu, D.R. Duh, Hamiltonicity of the pyramid network with or without fault,
Journal of Information Science and Engineering 25 (2009) 531–542.
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Lemma 2

A pyramid network PM[n] is hamiltonian
connected for n ≥ 1.
R.Y. Wu, D.R. Duh, Hamiltonicity of the pyramid network with or without fault,
Journal of Information Science and Engineering 25 (2009) 531–542.
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Lemma 3

A pyramid network PM[n] with one vertex
fault is hamiltonian for n ≥ 1.
R.Y. Wu, D.R. Duh, Hamiltonicity of the pyramid network with or without fault,
Journal of Information Science and Engineering 25 (2009) 531–542.
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Lemma 4

A mesh network M(m,n) is hamiltonian
laceable except : (1) m = 2 and n ≠ 2, and (2)
m = 3 and n = 2m.
G. Simmons, Almost all n-dimensional rectangular lattices are Hamiltonian
laceable, Congressus Numerantium (1978) 103–108.
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Lemma 5

A mesh network M(m,n) is hamiltonian for
m = n.
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Theorem 1



Assume that n ≥ 1.
Let s and t be any two distinct vertices of a
pyramid network PM[n].
Then there exists a 3*-container C3*(s,t) of
PM[n].
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Proof of Theorem 1



We prove this theorem by induction.
By brute force, we check the theorem holds
for n = 1.
Assume the theorem holds for any PM[n-1]
with n > 1.
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Case 1: s = (k1; x, y) and t = (k2;w, z)
with 1 ≤ k1, k2 ≤n -1.
s
hypothesis
PM[n-1]
t
(n-1)-layer
n-th layer
Lemma 4
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Case 2.1: s = (n; x, y) and t = (n;w, z)
Lemma 2
PM[n-1]
(n-1)-layer
n-th layer
s
Lemma 5
t
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Case 2.2: s = (n; x, y) and t = (n;w, z)
Lemma 3
PM[n-1]
u
(n-1)-layer
n-th layer
s
t
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Case 3.1~Case 3.4: s = (k; x, y) and t = (n;w, z) with 1≤
k≤n-1 except t∊{(n;1,1); (n; 1, 2n); (n; 2n, 1); (n;2n, 2n)}.
s
hypothesis
PM[n-1]
(n-1)-layer t’
n-th layer
Algorithm
t
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Case 4: s = (k; x, y) and t ∊{(n;1,1); (n; 1, 2n); (n;
2n, 1); (n;2n, 2n)}.
Case 4.1: s ≠ t’
s
hypothesis
PM[n-1]
(n-1)-layer t’
n-th layer
Algorithm
t
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Case 4: s = (k; x, y) and t ∊{(n;1,1); (n; 1, 2n); (n;
2n, 1); (n;2n, 2n)}.
Case 4.2: s = t’
PM[n-1]
(n-1)-layer s
Lemma 3
n-th layer
t
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Corollary 1


The pyramid network PM[n] is super
connected for n ≥ 1.
By Lemma 1, Lemma 2, and Theorem 1, the
pyramid network is k*-connected for 1 ≤k ≤ 3.
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