Pointwise Clique-Safe Domination in the Complement and Complementary Prism of Special Families of 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. 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.
Keywords: Clique-safe domination, pointwise clique-safe domination number, clique centrality
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References
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