Pointwise Clique-Safe Domination in Graphs

Abstract

Let G = (V (G),E (G)) be any finite, undirected, simple graph. The clique centrality of a vertex \(\mathit{x}\) \(\in\) V (G), denoted by \(\omega\)G (\(\mathit{x}\)), is the maximum size of a clique in G containing \(\mathit{x}\). A set D \(\subseteq\) V (G) is introduced in this paper as a pointwise clique-safe dominating set of G if for every vertex \(\mathit{y}\) \(\in\) Dc there exists a vertex \(\mathit{x}\) \(\in\) D such that \(\mathit{x}\)\(\mathit{y}\) \(\in\) E (G) where \(\omega\)\(\small\langle\)D\(\small\rangle\)G (\(\mathit{x}\)) \(\ge\) \(\omega\)\(\small\langle\)Dc\(\small\rangle\)G (\(\mathit{y}\)). The smallest cardinality of such a pointwise clique-safe dominating set of G is called the pointwise clique-safe domination number of G, denoted by \(\gamma\)pcs (G). This study aims to generate some observable properties of the parameter and to evaluate the minimum pointwise clique-safe dominating sets of some special families of graphs such as the complete graph K\(\mathit{n}\), fan graph F\(\mathit{n}\), wheel graph W\(\mathit{n}\) and complete bipartite K\(\mathit{m}\),\(\mathit{n}\) as well as graphs obtained under the mycielski operation.