Introducing the Clique-Safe Domination in Graphs

LetG= (V(G),E(G)) be any finite, undirected, simple graph. A setD⊆V(G) is introduced in this paperas a clique-safe dominating set ofGifDis a dominating set ofGand for every cliqueD′mof sizeminthe subgraph induced byV(G)rD, there exists a cliqueDnof sizenin the subgraph induced byDsuchthatn≥m. The clique-safe domination number ofG, denoted byγcs(G), is the smallest cardinality of aclique-safe dominating set ofG. This study aims to generate a few elementary properties of the parameterand to characterize the minimum clique-safe dominating sets of paths and cycles. As a consequence, theclique-safe domination numbers of the aforesaid graphs are obtained.