Introducing the Clique-Safe Domination in Graphs

Abstract

Let G = (V (G), E(G)) be any finite, undirected, simple graph. A set D \(\subseteq\) V (G) is introduced in this paper as a clique-safe dominating set of G if D is a dominating set of G and for every clique D\(\prime\)m of size m in the subgraph induced by V (G) \D, there exists a clique Dn of size n in the subgraph induced by D such that n \(\ge\) m. The clique-safe domination number of G, denoted by \(\gamma\)cs (G) is the smallest cardinality of a clique-safe dominating set of G. This study aims to generate a few elementary properties of the parameter and to characterize the minimum clique-safe dominating sets of paths and cycles. As a consequence, the clique-safe domination numbers of the aforesaid graphs are obtained.