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 $\gamma_k(G) \geq \gamma(G)+k-2$ where $ \Delta(G)\geq k\geq 2$ and this bound is sharp for every $k\geq 2$. In this paper, we characterize bipartite graphs satisfying the equality for $k\geq 3$ 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.