On \(k\)-Connected Restrained Domination in Graphs

Abstract

In this paper, we initiate the study of \(k\)-connected restrained domination in graphs. Let \(G = (V,E)\) be a graph. A \(k\)-connected restrained dominating set is a set \(S \subseteq V\) where \(S\) is a restrained dominating set and \(G[S]\) has at most \(k\) components. The \(k\)-connected restrained domination number of \(G\), denoted by \(\gamma_r^k(G)\), is the smallest cardinality of a \(k\)-connected restrained dominating set of \(G\). First, some exact values and sharp bounds for \(\gamma_r^k(G)\) are given in Section 2. Then, the necessary and sufficient conditions for \(\gamma_r(G) = \gamma_r^1(G) = \gamma_r^2(G)\) are given if \(G\) is a tree or a unicyclic graph in Section 3 and Section 4.