International Journal of Pure and Applied Mathematics
Volume 101 No. 6 2015, 985-991
ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)
url: http://www.ijpam.eu
AP
ijpam.eu
TOTAL BONDAGE NUMBER OF
GRID-BASED PYRAMID NETWORKS
Jasintha Quadras1 , Indra Rajasingh2 , A. Sajiya Merlin Mahizl3
1,3 Department
of Mathematics
Stella Maris College
Chennai 600 086, INDIA
2 School of Advanced Sciences
VIT University
Chennai, 600 127, INDIA
Abstract: A set D of a vertices in a graph G = (V, E) is said to be a total
dominating set of G if every vertex in V is adjacent to some vertex in D. The
total domination number of a graph G without isolated vertices is the minimum
cardinality of a total dominating set. The total bondage number bt (G) of G
is the minimum number of edges whose removal enlarges the total domination
number. In this paper we determine the exact values of the total domination
number and the total bondage number for mesh pyramid and W K-recursive
mesh pyramid.
AMS Subject Classification: 05C69
Key Words: interconnection networks, mesh, W K-recursive mesh, pyramid,
total dominating set, total domination number, total bondage number
1. Introduction and Preliminaries
The total domination in graphs was introduced by Cockayne et al. [2] in 1980.
The total domination in graphs has been extensively studied in the literature
[6, 7]. In 2009, Henning [8] gave a survey of selected recent results on this topic.
Received:
March 12, 2015
c 2015 Academic Publications, Ltd.
url: www.acadpubl.eu
986
J. Quadras, I. Rajasingh, A.S.M. Mahizl
In 1990, Fink et al. [4] introduced the bondage number as a parameter
for measuring the vulnerability of the interconnection network under link failure.We consider only simple finite graphs without isolated vertices. To each
vertex v of a graph G, N (v) denotes the set of all vertices of G which are
adjacent to v. For any subset S ⊆ V, N (S) = ∪{N (v)|v ∈ S}.
A set D of vertices in a graph G is said to be a dominating set if every
vertex in V − D is adjacent to some vertex in D. We call D a total dominating
set for G if every vertex in V is adjacent to some vertex in D.(i.e.,N (D) = V ).
The minimum cardinality of a dominating set (a total dominating set) of G
is denoted by γ(G), (γt (G)) and is called the domination number (the total
domination number ) of G. It is clear that γ(G) ≤ γt (G) ≤ 2γ(G) for any graph
G without isolated vertices.
The bondage number b(G) of a nonempty graph G is the minimum number
of edges whose removal from G results in a graph with larger domination number
than γ(G).
Following Fink et al. [4], Kulli and Patwari [12] proposed the concept of the
total bondage number for a graph. The total bondage number bt (G) of a graph
G is the minimum number of edges whose removal results in a graph with total
domination number larger than γt (G).
Kulli and Patwari [12] calculated the exact values of bt (G) for some standard
graphs such as a cycle Cn and a path Pn for n ≥ 4, a complete bipartite graph
Km,n and a complete graph Kn . Sridharan et al. [15], showed that for any
positive integer k there exists a tree T with bt (T ) = k. Fu-Tao et al. [10],
determined the exact values of bt (Gn,2 ), bt (Gn,3 ) and established some upper
bounds of bt (Gn,4 ). Further they derived upper bounds for bt (G) of a graph G
in terms of order. In [11], Hu and Xu showed that the problem of determining
total bondage number for general graphs is NP-hard.
An interconnection network can be modeled by an undirected graph in
which a processor is represented by a node(vertex) and a communication channel between two nodes is represented by an edge between the corresponding
nodes. The mesh, torus, W K-recursive mesh and hypermesh are examples of
grid-based interconnection network topologies. Many efficient algorithms for
these networks have been developed and their important topological properties
have been reported in the literature [14, 1, 3, 13, 5].
The conventional pyramid network (a mesh-pyramid) is one of the important network topologies as it has been used in both hardware architectures
and software structures for parallel computing, graph theory, digital geometry,
machine vision, and image processing [1, 3].
While the definition of the standard pyramid network is based on the 2-D
TOTAL BONDAGE NUMBER OF...
987
mesh topology for interconnecting the nodes in each level of its hierarchical
structure, other topologies can be employed for this purpose. In this paper
we determine the exact values of the total domination number and the total
bondage number for mesh pyramid and W K-recursive mesh pyramid.
2. The Grid-Based Pyramid
Definition 2.1. An a × b mesh, Ma,b , is a set of nodes V (Ma,b ) =
{(x, y)|1 ≤ x ≤ a, 1 ≤ y ≤ b} where nodes (x1 , y1 ) and (x2 , y2 ) are connected
by an edge iff |x1 − x2 | + |y1 − y2 | = 1 [9].
Definition 2.2. An L-level W K-recursive network based on 4-tuple nodes,
denoted by W K(4,L) , consists of a set of nodes V (W K(4,L) ) = {aL aL−1 . . . a1 |0 ≤
ai < 4}. The node with address schema A = {aL aL−1 . . . a1 } is connected to 1)
all the nodes with addresses (aL aL−1 . . . a2 k) that 0 ≤ k < 4, k 6= a1 , as sisters
nodes and 2) node (aL aL−1 . . . aj+1 aj−1 (aj )j ) if for one j, 1 ≤ j < 4; aj−1 =
aj−2 = aj−3 = a1 and aj 6= aj−1 , as cousin node. Notation (aj )j ) denotes j
consecutive a′j s. [9].
Figure1: (a) Mesh M4×4 (b) W K-recursive mesh W K(4,2)
Definition 2.3. A grid based pyramid of n levels, denoted by PG,n , consists of a set of nodes V (PG,n ) = {(k, x, y)|0 ≤ k ≤ n, 1 ≤ x, y ≤ 2k }. A node
(k, x, y) ∈ V (PG,n ) is said to be a node at level k. All the nodes in level k
form a 2k × 2k grid network G, which can be one of the grid-based networks:
mesh, torus, hypermesh, or W K-recursive mesh (G ∈ {M, T, HM, W K}). The
resulted pyramids can then be denoted as PM,n , PT,n , PHM,n and PW K,n respectively [9].
988
J. Quadras, I. Rajasingh, A.S.M. Mahizl
There are a total of N =
n
P
4k = (4(n+1) − 1)/3 vertices in a PG,n . In PG,n ,
k=0
each three consecutive levels from the last level are called 3-level blocks. The
(n − i)th level vertices where i = 1, 4, 7 . . . (3⌊ n+1
3 ⌋ − 2) are called middle level
vertices.
Figure 2:(a) Mesh pyramid PM,2 (b) W K-recursive pyramid PW K,2
3. The Total Domination Number
The total domination number is the minimum cardinality of a total dominating
set of G.
The following results are trivial.
Lemma 3.1.
γt (PG,1 ) = 2.
If PG,1 , G ∈ {M, W K} is a grid-based pyramid, then
Lemma 3.2.
γt (PG,2 ) = 4.
If PG,2 , G ∈ {M, W K} is a grid-based pyramid, then
Theorem 3.3. If PG,n , G ∈ {M, W K} is a grid-based pyramid with n > 1,
⌊ n+1
⌋
3
then γt (PG,n ) ≥
P
2n−(3i−2) .
i=1
Proof. In each 3-level block, the middle level vertices dominate the vertices
in the above and below the middle level. By induction hypothesis, all the
TOTAL BONDAGE NUMBER OF...
989
middle level vertices of each 3-level block form a total dominating set for PG,n .
⌋
⌊ n+1
3
Therefore, γt (PG,n ) ≥
P
2n−(3i−2) .
i=1
Procedure TOTAL DOMINATION NUMBER
PG,n , G ∈ {M, W K}, n ≥ 4
Input: Mesh pyramid or W K-recursive pyramid with n ≥ 4
Algorithm:
1. If n ≡ 0 (mod 3), select all the vertices in (n − i)th level, where i =
1, 4, 7 . . . (n − 2) and n .
2. If n ≡ 1 (mod 3), select all the vertices in (n − i)th level, where i =
1, 4, 7 . . . (n − 3) and add the number 2.
3. If n ≡ 2 (mod 3), select all the vertices in (n − i)th level, where i =
1, 4, 7 . . . (n − 1).
Output: γt (PG,n ) = number of vertices in (n − 1)th level +γt (PG,n−3 ).
Proof of Correctness: Since the (n − 1)th level vertices are adjacent only
to nth level vertices, we take (n − 1)th level vertices. Moreover, (n − 2)th and
nth level vertices are dominated by (n − 1)th level vertices and (n − 5)th and
(n − 3)rd level vertices are dominated by (n − 4)th level vertices and so on.
By the algorithm, (n − i)th level vertices where i = 1, 4, 7...(n − 2) and any
one vertex of level 1 form a minimum total dominating set for n ≡ 0 (mod 3)
and (n − i)th level vertices, where i = 1, 4, 7 . . . (n − 1) form a minimum total
dominating set for n ≡ 2 (mod 3).
Since each level contains 4i vertices, where i = 0, 1, 2, 3, . . . , n and the total
domination number of mesh pyramid or W K-recursive pyramid of level 1 is 2,
the sum of vertices in (n − i)th level, where i = 1, 4, 7 . . . (n − 3) and any two
vertices of level 1 form a minimum total dominating set for n ≡ 1 (mod 3).
4. The Total Bondage Number
Definition 4.1. Let G be a graph. If there exists E0 ⊂ E(G)such that (i)
there is no isolated vertex in G − E0 and (ii) γt (G − E0 ) > γt (G), then the edge
set E0 is called a total bondage edge set for G. If there is at least one total
bondage edge set for G, we define bt (G) = min{| E0 |: E0 is a total bondage
990
J. Quadras, I. Rajasingh, A.S.M. Mahizl
edge set of G}. Otherwise we put bt (G) = ∞. For star graph K1,n , we define
bt (G) = ∞ [12].
Theorem 4.2. If PG,n , G ∈ M, W K is a grid-based pyramid with n ≥ 1,
then bt (PG,n ) = 1.
Proof. Removal of any edge between (n−i)th and (n−(i+1)th level vertices
where i = 0, 3, 6, 9, . . . , (n − 3), increases the total domination number of PG,n
for n ≡ 0 (mod 3).
Removal of any edge between (n − i)th and (n − (i + 1)th level vertices where
i = 0, 3, 6, 9, . . . , (n − 4), increases the total domination number of PG,n for
n ≡ 1 (mod 3).
Removal of any edge between (n − i)th and (n − (i + 1))th level vertices
where i = 0, 3, 6, 9, . . . , (n − 2), increases the total domination number of PG,n
for n ≡ 2 (mod 3). Hence, bt (PG,n ) = 1.
5. Conclusion
In this paper we determine the total domination number and the total bondage
number of mesh pyramid and W K-recursive mesh pyramid. The problem of
finding the total domination number and the total bondage number of architectures such as Spider Web networks and Circulant networks are under investigation.
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