Bipartite graphs with close domination and k-domination numbers

Abstract

Let $k$ be a positive integer and let $G$ be a graph with vertex set $V(G)$. A subset $D\subseteq V(G)$ is a $k$-dominating set if every vertex outside $D$ is adjacent to at least $k$ vertices in $D$. The $k$-domination number ${\gamma }_k(G)$ is the minimum cardinality of a $k$-dominating set in $G$. For any graph $G$, we know that ${\gamma }_k(G)\ge \gamma (G)+k-2$ where $\Delta (G)\ge k\ge 2$ and this bound is sharp for every $k\ge 2$. In this paper, we characterize bipartite graphs satisfying the equality for $k\ge 3$ and present a necessary and sufficient condition for a bipartite graph to satisfy the equality hereditarily when $k=3$. We also prove that the problem of deciding whether a graph satisfies the given equality is NP-hard in general.