Introducing the Clique-Safe Domination in Graphs
John Mark R. Liwat *
Department of Mathematics, College of Arts and Sciences, Central Mindanao University, University Town, Musuan, Maramag, Bukidnon, 8714, Philippines.
Rolito G. Eballe
Department of Mathematics, College of Arts and Sciences, Central Mindanao University, University Town, Musuan, Maramag, Bukidnon, 8714, Philippines.
*Author to whom correspondence should be addressed.
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.
Keywords: Clique-safe dominating set, clique-safe domination number, clique-safe, clique