Quest Journals
Journal of Research in Applied Mathematics
Volume 6 ~ Issue 1 (2020) pp: 01-03
ISSN(Online) : 2394-0743 ISSN (Print): 2394-0735
www.questjournals.org
*Corresponding Author : Ortiz. L 1 | Page
Research Paper
Unique Metro Domination of Square of Cycles
Kishori P. Narayankar
#1
, Denzil Jason Saldanha
#2
, John Sherra
#3
#1
Department of Mathematics, Mangalore University, Mangalagangothri, Mangalore-574199, India.
#2
Department of Mathematics, Mangalore University, Mangalagangothri, Mangalore-574199, India.
#3
Department of Mathematics (Retired), St Aloysius College (Autonomous), Mangalore-575003, India.
Abstract: A dominating set D of G which is also a resolving set of G is called a metro dominating set. A metro
dominating set D ofa graph G(V,E) is a unique metro dominating set (in short an UMD-set) if|N(v)∩D|=1for
each vertex ∈−andthe minimum cardinality of an UMD-set of G is the unique metro domination number
of G denoted by
. In this paper, wedetermine unique metro domination number of
graphs.
Keywords: Domination, metric dimension, metro domination, unique metro domination.
Received 07 Apr, 2020; Accepted 22 Apr, 2020 © The author(s) 2020.
Published with open access at www.questjournals.org
I. INTRODUCTION
All the graphs considered in this paper are simple, connected and undirected. The length of a shortest
path between two vertices and in a graph is called the distance between and and is denoted by (, ).
For a vertex of a graph, () denote the set of all vertices adjacent to and is called open neighborhood of .
Similarly, the closed neighborhood of is defined as = ∩ {}. Let (, ) be a graph. For each
ordered subset ={
1
,
2
,
3
, … ,
} of , each vertex ∈ can be associated with a vector of distances
denoted by Γ )=((
1
, ) ,
2
, , …
, . The set is said to be a resolving set of G, ifΓ/ ≠
Γ(), for every , ∈−. A resolving set of minimum cardinality is a and cardinality of a
metric basis is the of G. The k-tuple, Γ() associated to the vertex ∈ with respect to
a metric basis S, is referred as a code generated by S for that vertex . If Γ( )=(
1
,
2
,...,
), then
1
,
2
,
3
, … ,
are called components of the code of generated by and in particular
, 1 ≤≤ , is called
-component of the code of generated by .
A dominating set of a graph (, ) is the subset of having the property that for each vertex
∈ −, there exists a vertex ∈ such that is in . A dominating set of which is also a resolving
set of is called a . A metro dominating set of a graph (, ) is a
(in short an − ) if ∣ ∩ ∣ =1 for each vertex ∈ − and
the minimum of cardinalities of UMD-sets of G is the of denoted by
().
Consider
, ≥ 4 labelled as
1
,
2
, … ,
in anticlockwise direction. Join
to
+2
for 1 ≤ ≤ − 2,
−1
to
1
and
to
2
. The resulting graph is denoted by
2
.
Lemma 1: For any positive integer ,
2
≥
5
.
Proof: A vertex
dominates five vertices
,
−1
,
−2
,
+1
,
+2
.Therefore, if D is a minimal dominating set
then ∣∣ ≥
5
. Hence we have (
2
)≥⌈
5
⌉.
Definition 1. [8] Consider a set S with two or more vertices of the graph G and let
and
be twodistinct
vertices of S. Further, let and
′
denote two distinct
-paths in G. If either or
′
, say Pcontains only two
vertices of S namely
and
, then we refer to
and
as neighboring vertices of S.Then the set of all the
vertices of − {
} is called a gap of S determined by
and
and denoted by
,
if the following hold
1.
,
are neighboring vertices in S and
2. No vertex in S is adjacent to any vertex in − {
,
}.