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