Lower Bounds on the Distance Domination Number of a Graph

Abstract

For an integer $k\ge 1$, a (distance) $k$-dominating set of a connected graph $G$ is a set $S$ of vertices of $G$ such that every vertex of $V(G)\setminus S$ is at distance at most~$k$ from some vertex of $S$. The $k$-domination number, ${\gamma }_k(G)$, of $G$ is the minimum cardinality of a $k$-dominating set of $G$. In this paper, we establish lower bounds on the $k$-domination number of a graph in terms of its diameter, radius and girth. We prove that for connected graphs $G$ and $H$, ${\gamma }_k(G\times H)\ge {\gamma }_k(G)+{\gamma }_k(H)-1$, where $G\times H$ denotes the direct product of $G$ and $H$.