# Simultaneous Resolvability in Families of Corona Product Graphs

Research paper by **Yunior RamÃrez-;Cruz, Alejandro Estrada-;Moreno; Juan A. RodrÃguez-;VelÃ¡zquez**

Indexed on: **26 Aug '16**Published on: **19 Aug '16**Published in: **Bulletin of the Malaysian Mathematical Sciences Society**

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#### Abstract

Abstract
Let
\(\mathcal{G}\)
be a graph family defined on a common vertex set V and let d be a distance defined on every graph
\(G\in \mathcal{G}\)
. A set
\(S\subset V\)
is said to be a simultaneous metric generator for
\(\mathcal{G}\)
if for every
\(G\in \mathcal{G}\)
and every pair of different vertices
\(u,v\in V\)
there exists
\(s\in S\)
such that
\(d(s,u)\ne d(s,v)\)
. The simultaneous metric dimension of
\(\mathcal{G}\)
is the smallest integer k such that there is a simultaneous metric generator for
\(\mathcal{G}\)
of cardinality k. We study the simultaneous metric dimension of families composed by corona product graphs. Specifically, we focus on the case of two particular distances defined on every
\(G\in \mathcal{G}\)
, namely the geodesic distance
\(d_G\)
and the distance
\(d_{G,2}:V\times V\rightarrow \mathbb {N}\cup \{0\}\)
defined as
\(d_{G,2}(x,y)=\min \{d_{G}(x,y),2\}\)
.AbstractLet
\(\mathcal{G}\)
be a graph family defined on a common vertex set V and let d be a distance defined on every graph
\(G\in \mathcal{G}\)
. A set
\(S\subset V\)
is said to be a simultaneous metric generator for
\(\mathcal{G}\)
if for every
\(G\in \mathcal{G}\)
and every pair of different vertices
\(u,v\in V\)
there exists
\(s\in S\)
such that
\(d(s,u)\ne d(s,v)\)
. The simultaneous metric dimension of
\(\mathcal{G}\)
is the smallest integer k such that there is a simultaneous metric generator for
\(\mathcal{G}\)
of cardinality k. We study the simultaneous metric dimension of families composed by corona product graphs. Specifically, we focus on the case of two particular distances defined on every
\(G\in \mathcal{G}\)
, namely the geodesic distance
\(d_G\)
and the distance
\(d_{G,2}:V\times V\rightarrow \mathbb {N}\cup \{0\}\)
defined as
\(d_{G,2}(x,y)=\min \{d_{G}(x,y),2\}\)
.
\(\mathcal{G}\)
\(\mathcal{G}\)Vd
\(G\in \mathcal{G}\)
\(G\in \mathcal{G}\)
\(S\subset V\)
\(S\subset V\)
\(\mathcal{G}\)
\(\mathcal{G}\)
\(G\in \mathcal{G}\)
\(G\in \mathcal{G}\)
\(u,v\in V\)
\(u,v\in V\)
\(s\in S\)
\(s\in S\)
\(d(s,u)\ne d(s,v)\)
\(d(s,u)\ne d(s,v)\)
\(\mathcal{G}\)
\(\mathcal{G}\)k
\(\mathcal{G}\)
\(\mathcal{G}\)k
\(G\in \mathcal{G}\)
\(G\in \mathcal{G}\)
\(d_G\)
\(d_G\)
\(d_{G,2}:V\times V\rightarrow \mathbb {N}\cup \{0\}\)
\(d_{G,2}:V\times V\rightarrow \mathbb {N}\cup \{0\}\)
\(d_{G,2}(x,y)=\min \{d_{G}(x,y),2\}\)
\(d_{G,2}(x,y)=\min \{d_{G}(x,y),2\}\)