A Note on Weakly Connected Domination Number in Graphs

Abstract

Let \(G\) be a connected graph. A weakly connected dominating set of \(G\) is a dominating set \(D\) such that the edges not incident to any vertex in \(D\) do not separate the graph \(G\). In this paper, we first consider the relationship between weakly connected domination number \(\gamma_w(G)\) and the irredundance number \(ir(G)\). We prove that \(\gamma_w(G) \leq \frac{5}{2}ir(G) – 2\) and this bound is sharp. Furthermore, for a tree \(T\), we give a sufficient and necessary condition for \(\gamma_c(T) = \gamma_w(T) + k\), where \(\gamma_c(T)\) is the connected domination number and \(0 \leq k \leq \gamma_w(T) – 1\).