A note on a relation between the weak and strong domination numbers of a graph

Abstract

In a graph $G=(V,E)$ a vertex is said to dominate itself and all its neighbors. A set $D \subset V$ is a weak (strong, respectively) dominating set of $G$ if every vertex $v \in V-S$ is adjacent to a vertex $u \in D$ such that $d_G(v) \geq d_G(u)$ $d_G(v) \leq d_G(u)$, respectively). The weak (strong, respectively) domination number of $G$, denoted by $\gamma_w(G)$ $(\gamma_s(G)$, respectively), is the minimum cardinality of a weak (strong, respectively) dominating set of $G$. In this note we show that if $G$ is a connected graph of order $n \geq 3$, then $\gamma_w(G) + t\gamma_s(G) \leq n$, where $t=3/(\Delta+1)$ if $G$ is an arbitrary graph, $t=3/5$ if $G$ is a block graph, and $t=2/3$ if $G$ is a claw free graph.