Pointwise Clique-Safe Domination in the Complement and Complementary Prism of Special Families of Graphs

Abstract

Let G = (V (G), E(G)) be any finite, undirected, simple graph. The maximun size of a clique containing a vertex \(\mathit{x}\) \(\in\) V (G) is called the clique centrality of \(\mathit{x}\) , denoted by \(\omega\)G (\(\mathit{x}\)) . A set D \(\subseteq\) V (G) is said to be 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 \(\mathit{xy}\) \(\in\) E (G) where \(\omega\)\(\small\langle\)D\(\small\rangle\)G (\(\mathit{x}\)) \(\ge\) \(\omega\)\(\small\langle\)Dc\(\small\rangle\)G (\(\mathit{y}\)) . The smallest obtainable cardinality of a pointwise clique-safe dominating set of G is called the pointwise clique-safe domination number of G, denoted by \(\gamma\)\(\mathit{pcs}\) (G) . This study aims to generate some properties of the parameter and to characterize the minimum pointwise cliquesafe dominating sets of the complement of some special families of graphs as well as their complementary prism.