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Simultaneous Resolvability in Families of Corona Product Graphs

Research paper by Yunior Ramírez-;Cruz, Alejandro Estrada-;Moreno; Juan A. Rodríguez-;Velázquez

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



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 \(G\in \mathcal{G}\) \(G\in \mathcal{G}\) \(d_G\) \(d_G\) \(d_{G,2}:V\times V\rightarrow \mathbb {N}\cup \{0\}\) \(d_{G,2}:V\times V\rightarrow \mathbb {N}\cup \{0\}\) \(d_{G,2}(x,y)=\min \{d_{G}(x,y),2\}\) \(d_{G,2}(x,y)=\min \{d_{G}(x,y),2\}\)