On the outer-connected reinforcement and bondage problems in bipartite graphs: the algorithmic complexity

An outer connected dominating(OCD) set of a graph G=(V,E) is a set D~⊆V such that every vertex not in S is adjacent to a vertex in S, and the induced subgraph of G by V∖D~, i.e. G[V∖D~], is connected. The OCD number of G is the smallest cardinality of an OCD set of G. The outer-connected bondage number of a nonempty graph G is the smallest number of edges whose removal from G results in a graph with a larger OCD number. Also, the outer-connected reinforcement number of G is the smallest number of edges whose addition to G results in a graph with a smaller OCD number. In 2018, Hashemi et al. demonstrated that the decision problems for the Outer-Connected Bondage and the Outer-Connected Reinforcement numbers are all NP-hard in general graphs. In this paper, we improve these results and show their hardness for bipartite graphs. Also, we obtain bounds for the outer-connected bondage number.