Pointwise Clique-Safe Domination in Graphs

Let G = (V (G),E (G)) be any finite, undirected, simple graph. The clique centrality of a vertex V (G), denoted by
G (), is the maximum size of a clique in G containing
. A set D V (G) is introduced in this paper as a pointwise clique-safe dominating set of G if for every vertex Dc there exists a vertex D such that E (G) where D G () Dc G (). 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 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, fan graph F, wheel graph W and complete bipartite K
,as well as graphs obtained under the mycielski operation.