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