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 ${\gamma }_w(G)\le \frac{5}{2}ir(G)–2$ 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 $0\le k\le {\gamma }_w(T)–1$.