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 ⊆ 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 γ k (G) is the minimum cardinality of a k-dominating set in G. For any graph G, we know that γ k (G) ≥ γ(G) + k − 2 where ∆(G) ≥ k ≥ 2 and this bound is sharp for every k ≥ 2. In this paper, we characterize bipartite graphs satisfying the equality for k ≥ 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.