Secure Restrained Domination in Graphs

Abstract

Let G = (V, E) be a graph and let ${S \subseteq V}$. The set S is a dominating set of G if every vertex in ${V \backslash S}$ is adjacent to some vertex in S. The set S is a secure dominating set of G if for each ${u \in V \backslash S}$, there exists a vertex ${v \in S}$ such that ${uv \in E}$ and ${(S \backslash \{v\}) \cup \{u\}}$ is a dominating set of G. The set S is a restrained dominating set if every vertex in ${V \backslash S}$ is adjacent to a vertex in S and to a vertex in ${V \backslash S}$. A set ${S \subseteq V(G)}$ is called a secure restrained dominating set (SRDS) of G if S is restrained dominating and for all ${u \in V \backslash S}$ there exists ${v \in S \cap N(u)}$ such that ${(S \backslash \{v\}) \cup \{u\}}$ is restrained dominating. The minimum cardinality of a SRDS is called the secure restrained domination number of G and is denoted by ${\gamma_{sr}(G)}$. In this paper we study few properties of secure restrained domination number on certain classes of graphs and we evaluate ${\gamma_{sr}(G)}$ values for trees, unicyclic graphs, split graphs and generalized Petersen graphs.