International Journal of Recent Technology and Engineering (IJRTE)
ISSN: 2277-3878, Volume-8 Issue-3, September 2019
1003
Published By:
Blue Eyes Intelligence Engineering
& Sciences Publication
Retrieval Number: C4062098319/19©BEIESP
DOI:10.35940/ijrte.C4062.098319
Abstract: A collection = {H
1
,H
2
,..., H
r
} of induced sub
graphs of a graph G is said to be sg-independent if (i)
V(H
i
)V(H
j
)= , i j, 1≤ i, j≤ r and (ii) no edge of G has its
one end in H
i
and the other end in H
j
, i j, 1≤ i, j≤ r. If H
i
H,
∀ i, 1≤ i ≤r, then is referred to as a H-independent set of G. Let
be a perfect or almost perfect H-packing of a graph G. Finding
a partition
of such that
is H-
independent set, ∀ i,
1 ≤ i ≤ k, with minimum k is called the induced H-packing k-
partition problem of G. The induced H-packing k-partition
number denoted by ipp(G,H) is defined as ipp(G,H) = min
(G,H) where the minimum is taken over all H-packing of G. In
this paper we obtain the induced H-packing k-partition number
for Enhanced hypercube, Augmented Cubes and Crossed Cube
networks where H is isomorphic to
and
.
Keywords: Augmented Cubes, Crossed Cube Networks,
Enhanced hypercube, Induced H-packing k-partition.
I. INTRODUCTION
For any graph G, let V(G) denote the set of vertices in G
and E(G) denote the set of edges in G, |V(G)| and |E(G)|
denote the respective cardinalities of these sets. An H-
packing of a graph G = (V, E) is a set of vertex disjoint sub
graphs of G, each of which is isomorphic to a fixed graph H.
A perfect H-packing in a graph G is a set of H-subgraphs of
G such that every vertex in G is incident with one H-
subgraph in this set. An almost perfect H-packing in a graph
G is a set of H-subgraphs of G such that at most |V(H)| − 1
number of vertices are not incident on any H - subgraph in
G [13], [14]. We define this concept as follows: A collection
= {H
1
, H
2
, ... , H
r
} of induced sub graphs of a graph G is
said to be sg-independent if (i) V(H
i
)V(H
j
) = , i j, 1≤ i,
j≤ r and (ii) no edge of G has its one end in H
i
and the other
end in H
j
, i j, 1≤ i, j≤ r. If H
i
H, ∀ i, 1≤ i ≤ r, then is
referred to as a H-independent set of G. Let be a perfect
or almost perfect H-packing of a graph G. Finding a
partition
of such that
is H-
independent set, ∀i, 1≤ i ≤ k, with minimum k is called the
induced H-packing k-partition problem of G. The minimum
induced H-packing k-partition number is denoted by
ipp
H
(G,H). The induced H- packing k-partition number
denoted by ipp(G, H) is defined as ipp(G, H) = min
(G,
Revised Manuscript Received on September 15, 2019
* Correspondence Author
Santiagu Theresal
*
, Department of Mathematics, Loyola College,
University of Madras, Chennai - 034,India.Email:
santhia.teresa@gmail.com
Antony Xavier, Department of Mathematics, Loyola College,
University of Madras, Chennai - 034, India.
Email:dantonyxavierlc@gmail.com
S. Maria Jesu Raja, Department of Mathematics, Vels Institute of
Science, Technology and Advanced Studies, Chennai -117, India.
Email: jesur2853@gmail.com
H) where the minimum is taken over all H-packing of G.
Packing is an extension of matching. An Induced matching
and induced matching partitions of certain interconnection
networks was studied [2], [11]. The induced H-packing k-
partition problem was studied for certain interconnection
networks such as hypercubes, Sierpiński graphs [12]. An
approximation algorithm for maximum
-packing in
subcubic graphs was studied by Kosowski et al [10]. Xavier
et al [12] proved that the induced
-packing k-partition
problem is NP- complete, also induced
-packing k-
partition problem is NP-complete. In this paper we obtain
the induced H-packing k-partition number for Enhanced
hypercube, Augmented Cubes and Crossed Cube networks
where H is isomorphic to
and
.
II. ENHANCED HYPERCUBE NETWORKS
A Hypercube with extra connections called skips is referred
to as an enhanced hypercube. As a variant of the
,
enhanced hypercubes
(n ≥2, (1≤k ≤ n−1) are proposed
to improve the efficiency of the hypercube architecture and
have found substantial applications. Inherited from
,
is also a regular graph [15], [20]. But the enhanced
hypercubes are much more attractive than normal
hypercubes due to its potential nice topological properties.
The enhanced hypercube
(1≤k≤n−1), is a graph with
vertex set V(
) = V(
) and edge set E(
) = E(
)∪
(x
0
x
1
x
2
, ..., x
k−2
, x
k−1
, x
k
... x
n−1
, x
0
x
1
x
2
...x
k−2
, x
k−1
, x
k
...x
n−1
).
The edges of
in
are hypercube edges and the
remaining edges of
are called complementary edges
[4], [7], [8], [9], [16], [17], [18], [20], [21].When k=0,
reduces to the n-dimensional hypercube. The enhanced
hypercubes
(1≤ k ≤ n−1) proposed by Tzeng and Wei
[15] are (n +1) regular. They have 2
n
vertices and (n +1)2
n−1
edges.
Theorem 1.1. Let G be the Enhanced hypercube
network
n ≥2, Then G has an almost perfect
-packing.
Proof. We prove the result by induction on the dimension n
of the Enhanced hypercube network
. We begin with n =
2,
= {(00, 10, 11)} is an almost perfect
-packing
leaving out one vertex unsaturated. In
0
∪
is an almost perfect
-packing leaving out two
unsaturated vertices inducing an edge, where i
denotes the
set of paths in
prefixed by i, i = 0, 1. See Fig. 1(a).
Assume the result to be true for
. Consider
.
Suppose n + 1,2 is even. By induction hypothesis
=
0
∪1
is an almost perfect
-packing leaving out two
unsaturated vertices in each copy of
in
. By
construction the four left out
vertices induce a cycle C. Let P
be a sub path of length 2 in C.
Then
=0
∪1
∪ is
Induced H-Packing k-Partition Problem in Certain
Networks
Santiagu Theresal, Antony Xavier, S. Maria Jesu Raja