VISTAS Vol. 3, No. 1, 2014, pp. 107-112
ISSN: 2319-5770
PRECLUSION NUMBER FOR
TRIANGULAR PYRAMID
D. Antony Xavier1 and Deeni C.J.2
of Mathematics, Loyola College, Chennai, Tamilnadu–600034, India
2Racine Research Center, Loyola College, Chennai, Tamilnadu–600034, India
E-mail: [email protected]
1Department
Abstract—In this paper we use the concepts of matching, perfect matching and matching preclusion number.
The matching preclusion number of a graph G, denoted by mp(G), is the minimum number of edges whose
deletion leaves the resulting graphs without a perfect matching. A pyramid of n levels, denoted by Pn
consists of a set of vertices V(Pn) = {(k,x,y)/0 ≤ k ≤ n, 1 ≤ x,y ≤ 2k}. A Radix n-triangular mesh network
denoted as Tn, consists of a set of vertices V(Tn) = {(x,y)/ 0 ≤ x+y ≤ n}, where any two vertices (x1,y1) and
(x2,y2) are connected by an edge if | x1 - y1| + |x2 – y2|
n – 1. The number of vertices and edges in Tn
n
(
n

1
)
3
n
(
n

1
)
is equal to
and
respectively. A triangular pyramid(trippy) of n levels,TPn consists of
2
2
a set of vertices V(TPn) = {(k,(x,y))/1 ≤ k ≤ n, 0 ≤ x+y ≤ k} arranged of n levels of triangular mesh. Trippy
has n ( n  1) ( n  2 ) vertices and n(n+1)(n-1) edges. In this paper we try to find out the values of matching
6
preclusion number (mp(TPn)) forTPn when n ( n  1) ( n  2 ) is even and (mp(TPn)) is 3.
6
Keywords: Matching, Matching Preclusion Number, Perfect Matching, Triangular Pyramid
INTRODUCTION
In this paper we use only the finite simple undirected graph. i,e without loops or
multiple edges. Let G be a graph of order n, and also consider this n is even. A
matching M of G is a set of pair wise non-adjacent edges. A perfect matching in G is a
set of edges such that every vertex is incident with exactly one edge in the set. The
matching preclusion number of a graph G, denoted by mp (G), is the minimum
number of edges whose deletion leaves the resulting graph without a perfect
matching. The concept of matching preclusion was introduced by Birgham et al., [4]
and further studied by Cheng and Liptak [3], with special attention given to
interconnection networks. In [2] Park also put forward some results on perfect
matching. In [2], the matching preclusion number was determined for three classes of
graphs, namely, the complete graphs, the complete bipartite graphs Km,n and
the hypercube.
Interconnection networks are currently being used for many different applications
ranging from inter–ip connections in VLSI circuits to wide area computer networks [7].
An interconnection network can be modeled by a graph where a processor is
represented by a vertex, and a communication channel between two processing
vertices is represented by edge. Various topologies for interconnection networks have
been proposed in the literature: these include cubic networks (e.g. meshes, k-ary
n-cubes, hypercubes, folded cubes and hypermeshes), hierarchical networks
(pyramids and trees), and recursive networks (e.g., RTCC networks, OTIS networks,
WK recursive networks and star graphs) that have been widely studied in the
literature for topological properties [7].
A famous network topology being used as the base of both hardware architectures
and software structures is the pyramid. By exploiting the inherent hierarchy at each
level, pyramid structures can efficiently handle the communication requirements of
various problems in graph theory, digital geometry, machine vision, and image
processing [7].
108  VISTAS
The main problems with traditional pyramids are hardware scalability and poor
network connectivity and bandwidth. To address these problems, in paper [6] they
propose a new pyramidal network, called triangular pyramid which is based on
triangular mesh instead of the traditional two dimensional (2D) mesh employed by
traditional pyramids. The new network preserves many desirable properties of
traditional pyramid network.
Proposition 1.1
Let G be a graph with an even number of vertices. Then mp (G) ≤ δ (G), where δ (G)
is the minimum degree of G.
Proof:
Deleting all edges incident to a single vertex will give a graph with no perfect
matching and the result follows.
TRIANGULAR MESH AND TRIANGULAR PYRAMID
In this section, we give the definitions and basic properties of triangular mesh and
triangular pyramid (Trippy).
Definition 2.1 [5] An a × b mesh, Ma,b is a set of vertices V(Ma,b) = {(x, y)/1 ≤ x ≤ a,
1 ≤ y ≤ b} where any two vertices (x1,y1) and (x2,y2)are connected by an edge if and only
if | x1  y1| + |x2  y2| = 1.
Definition 2.2 [5] A pyramid of n levels, denoted by Pn consists of a set of vertices
V(Pn) = {(k, x, y)/ 0 ≤ k ≤ n, 1 ≤ x, y ≤ 2k}. A vertex labeled (k, x, y) ∈ V(Pn) is said to be a
vertex in level k. All the vertex in level k form a 2k×2k mesh network. There are a total
of N = ∑
4 =
vertices.
Definition 2.3 [7] A radix n-triangular mesh network, denoted as Tn, consists of a
set of vertices V(Tn) ={ (x, y)/ 0 ≤ x + y ≤ n} where any two vertices (x1,y1) and (x2, y2)
are connected by an edge if and only if │x1  x2│+ │y1  y2│ = n – 1. The number of
vertices and edges in Tn is equal to n(n + 1)/2 and 3n(n  1)/2, respectively.
Example T6
Definition 2.4 [7]
A Trippy of n levels (or layers), denoted as TPn consists of a set of vertices V(TPn) =
{(k, (x, y))/ 1 ≤ k ≤ n, 0 ≤ x + y ≤ k} arranged of n levels of triangular mesh. A vertex is
addressed as (k, (x, y)) and is said to be a vertex at level k. The part (x, y) of the
address determines the address of a vertex within the layer k, of the triangular
network. The vertices at level k form a network Tk. Hence there exist a total of
VISTAS  109
(
)(
)
vertices and n(n + 1)(n  1) edges. A vertex with address (k, (x, y)) placed
at level k, of the Tk network, is connected to adjacent vertices as defined in the above
definition. This vertex is also connected to the vertices (x, y), (x + 1, y), (x, y + 1) in
level k + 1.
Example
Fig. 2
Lemma 2.1 [7] Any Radix triangular mesh network is Tn is Hamiltonian.
Lemma 2.2 [1] Let n be an integer and n(n + 1) ≡ 0 (mod 4) then Tn has exactly two edge disjoint
perfect matchings.
Theorem 2.1 [1] Let n > 3 be an integer and n(n + 1) ≡ 0 (mod 4), then the following statements
hold.

Mp (Tn) = 2

Every minimum matching preclusion set in Tn is trivial.
MATCHING PRECLUSION NUMBER FOR TRIANGULAR PYRAMID
In this section we find out the matching preclusion number for the triangular pyramid
TPn
Remark 1 Let n be an integer and n(n+1)(n+2) is even. Then TPn has exactly three
edge disjoint perfect matchings.
Proposition 3.1 Consider TPn where n ≥ 3 be an integer and
ve(G) = 6.
Theorem 3.1 Let n ≥ 3 be an integer and
statements hold.
(
)(
)
(
)(
)
is even. Then
is even, then the following
1. mp(TPn) = 3
2. Every minimum matching preclusion set in TPn is trivial.
Proof:
1. By the remark stated above TPn has exactly three edge disjoint perfect
matching M1, M2, M3∴ mp(TPn) > 2
110  VISTAS
But mp(TPn) ≤ δ = 3 by prop 1.1
∴ mp(TPn) = 3
2. Consider two perfect matching M1 and M2 of TPn
Let F be a preclusion set in TPn and let F = { x,y,z}.
Clearly | F | = 3
TPn – F has a perfect matching.
Case I
Let x, y, z ∈ M1
Then M2 ⊆ TPn – F
∴ TPn – F has a perfect matching.
Similarly x, y, z ∈ M2
Case II
1. Let x ∈ M1, y, z ∈ TPn – M1 and y, z not in the parellogram.
Suppose y, z is an interior edge of TPn.
Clearly opposite sides of the parallelogram is also in M1.
Fig. 3
Suppose x = {(k, (i, j)), (k, (I + 1, j))}.
Then {(k, (I 1, j + 1)), (k, (i, j + 1))} ∈ M1
Now, M1 + {(k, (i, j)), (k, (I + 1, j + 1))}, {(k, (I + 1, j)), (k, (i, j + 1))} – {x, {(k, (I  1, j +
1)), (k, (i, j + 1))}}
is a perfect matching in TPn – F.
2. Let x ∈ M1 and y, z∉ M1
Fig. 4
Suppose x = {(k, (i, j + 1)), (k, (i + 1, j + 1))}.
Then {(k, (i+1, j)), (k, (i+2, j))}, {(k, (i, j)), (k, (i-1, j+1))} ∈ M1
Now, M1 + {(k, (i - 1, j+ 1)), (k, (i, j + 1))}, {(k, (i, j)), (k, (i + 1, j))}, {(k, (i+1, j + 1)), (k, (i
+ 2, j))} - {x, {(k, (i, j)), (k, (i - 1, j + 1))}, {(k, (i + 1, j)), (k, (i + 2, j))}} is a perfect matching.
VISTAS  111
Case III
Let x, y∈ M1 and z ∉ M1
Fig. 5
Suppose x = { (k, (i, j + 1)), (k, (i + 1, j + 1))} and y = {(k, (i + 1, j)), (k, (i + 2, j))}.
Then { (k, (i, j)), (k, (i – 1, j + 1))}∈ M1
Now M1 + {(k, (i – 1, j + 1)), (k, (i, j + 1))}, {(k, (i, j)), (k, (i + 1, j))}, {(k, (i + 1, j + 1)), (k,
(i + 2, j))} – {x, y, {(k, (i, j)), (k, (i – 1, j + 1))}} is a perfect matching.
2. Let x, y∈ M1 and z ∉ M1 also x, y, z are continuous edges.
Fig. 6
Suppose x = {(k, (i, j)), (k, (i + 1, j))} and y = { (k, (i + 2, j)), (k, (i + 3, j))}.
Then {(k, (i – 1, j + 1)), (k, (i, j + 1))}, {(k, (i + 1, j + 1)), (k, (i + 2, j + 1))} ∈ M1
Now M1 + {(k, (i, j)), (k, (i – 1, j + 1))}, {(k, (i + 1, j)), (k, (i, j + 1))}, {(k, (i + 2, j)), (k, (i +
1, j + 1))}, {(k, (i + 3, j)), (k, (i + 2, j + 1))} – {x, y, {(k, (i – 1, j + 1)), (k, (i, j + 1))}, {(k, (i + 1,
j + 1)), (k, (i + 2, j + 1))}} is a perfect matching.
3. Let x∈ M1 and y, z ∉ M1 also x, y, z are continuous edges.
Fig. 7
Suppose x = {(k, (i + 1, j)), (k, (i + 2, j))}.
Then {(k, (i, j)) (k, (i – 1, j + 1))}, {(k, (i, j + 1))(k, (i + 1, j + 1))}, {(k, (i + 3, j)), (k, (i + 2,
j + 1))} ∈ M1
Now M1 + {(k, (i, j)), (k, (i – 1, j + 1))}, {(k, (i + 1, j)), (k, (i, j + 1))}, {(k, (i + 2, j)), (k, (i +
1, j + 1))}, {(k, (i + 3, j)), (k, (i + 2, j + 1))} – {x, {(k, (i, j)) (k, (i – 1, j + 1))}, {(k, (i, j + 1))(k, (i
+ 1, j + 1))}, {(k, (i + 3, j)), (k, (i + 2, j + 1))}} is a perfect matching.
CASE IV Let x ∈ M1 and y, z∉ M1 also x, y, z are incident on single vertex.
Suppose x ={(k, (i – 1, j + 1)), (k, (i, j + 1))}
112  VISTAS
Fig. 8
Also {(k, (i, j)), (k, (i + 1, j))} ∈ M1
Now M1 + {(k, (i, j)), (k, (i – 1, j + 1))}, {(k, (i, j)), (k, (i, j + 1))} – {x, {(k, (i, j)), (k, (i + 1,
j))}} is a perfect matching.
Suppose x ={(k, (i, j + 1)), (k, (i + 1, j + 1))}
(k,(i-1,j+1)) (k,(i,j+1)) (k,(i+1,j+1))
(k,(i-1,j+1)) (k,(i,j+1)) (k,(i+1,j+1))
(k,(i,j))
(k,(i+1,j))
(k,(i+2,j))
(k,(i,j))
(k,(i+1,j)) (k,(i+2,j))
Fig. 9
Then {(k, (i, j)) (k, (i – 1, j + 1))}, {(k, (i + 1, j))(k, (i + 2, j))} ∈ M1
Now M1 + {(k, (i – 1, j + 1)), (k, (i, j + 1))}, {(k, (i, j)), (k, (i + 1, j))}, {(k, (i + 2, j)), (k, (i +
1, j + 1))} – {x, {(k, (i, j)) (k, (i – 1, j + 1))}, {(k, (i + 1, j))(k, (i + 2, j))}} is a perfect matching.
Suppose x ={(k, (i + 2, j)), (k, (i + 1, j + 1))}
Fig. 10
Then {(k, (i, j)) (k, (i + 1, j))}, {(k, (i – 1, j + 1))(k, (i, j + 1))} ∈ M1
Now M1 + {(k, (i, j)), (k, (i – 1, j + 1))}, {(k, (i, j + 1)), (k, (i + 1, j))}, {(k, (i + 1, j)), (k,
(i + 2, j))} – {x, {(k, (i – 1, j + 1)) (k, (i, j + 1))}, {(k, (i, j))(k, (i + 1, j))}} is a perfect matching.
ACKNOWLEDGEMENT
This work is supported by Maulana Azad Fellowship F1–17.1/2013-14/MANF–CHR–
KER–30436 of the University Grants Commission, New Delhi, India.
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