Metro Domination of Square Cycle

Abstract

Let $G=(V,E)$ be a graph.A set $S\subseteq V$ is called resolving set if for every $u,v\in V$ there exist $w\in V$, such that $d(u,w)\neq d(v,w)$. The resolving set with minimum cardinality is called metric basis and its cardinality is called metric dimention and it is denoted by $\beta (G)$. A set $D\subseteq V$ is called dominating set if every vertex not in $D$ is adjacent to at least one vertex in $D$. The dominating set with minimum cardinality is called domination number of $G$ and it is denoted by $\gamma (G)$. A set which is both resolving set as well as dominating set is called metro dominating set. The minimum cardinality of a metro dominating set is called metro domination number of $G$ and it is denoted by $\gamma_ \beta(G)$. In this paper we determine metro domination number of square cycle.