The algorithmic complexity of bondage and reinforcement problems in bipartite graphs

Let G=(V,E)G=(V,E) be a graph. A subset D⊆VD⊆V is a dominating set if every vertex not in D is adjacent to a vertex in D. The domination number of G  , denoted by γ(G)γ(G), is the smallest cardinality of a dominating set of G. The bondage number of a nonempty graph G is the smallest number of edges whose removal from G   results in a graph with domination number larger than γ(G)γ(G). The reinforcement number of G is the smallest number of edges whose addition to G   results in a graph with smaller domination number than γ(G)γ(G). In 2012, Hu and Xu proved that the decision problems for the bondage, the total bondage, the reinforcement and the total reinforcement numbers are all NP-hard in general graphs. In this paper, we improve these results to bipartite graphs.