Global Secure Domination in Graphs

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.