Global Secure Domination in Graphs

Abstract

Let $G=(V,E)$ be a graph. A subset S of V is called a dominating set of G if every vertex in $V\backslash S$ is adjacent to a vertex in S. A dominating set S is called a secure dominating set if for every vertex $v\in V-S,$ there exists $u\in S$ such that $uv\in E$ and $(S-\{u\})\cup \{v\}$ is a dominating set of G. If S is a secure dominating set of both G and its complement $\overline{G},$ then S is called a global secure dominating set (gsd-set) of G. The minimum cardinality of a gsd-set of G is called the global secure domination number of G and is denoted by $\gamma _{gs}(G).$ In this paper we present several basic results on $\gamma _{gs}(G)$ and interesting problems for further investigation.