Lower Bounds on the Distance Domination Number of a Graph

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, $γ_k(G)$, 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 $H$, $γ_k(G \times H) \ge γ_k(G) + γ_k(H) -1$, where $G \times H$ denotes the direct product of $G$ and $H$.

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