On the complexity of the outer-connected bondage and the outer-connected reinforcement problems

Let G = (V,E) be a graph. A subset S ⊆ V is a dominating set of G if every vertex not in S is adjacent to a vertex in S. A set D̃ ⊆ V of a graph G = (V,E) is called an outer-connected dominating set for G if (1) D̃ is a dominating set for G, and (2) G[V \ D̃], the induced subgraph of G by V \ D̃, is connected. The minimum size among all outer-connected dominating sets of G is called the outerconnected domination number of G and is denoted by γ̃c(G). We define the outer-connected bondage number of a graph G as the minimum number of edges whose removal from G results in a graph with an outer-connected domination number larger than the one for G. Also, the outer-connected reinforcement number of a graph G is defined as the minimum number of edges whose addition to G results in a graph with an outer-connected domination number, which is smaller than the one for G. This paper shows that the decision problems for the outer-connected bondage and the outer-connected reinforcement numbers are NP-hard. Also, the exact values of the bondage number are determined for several classes of graphs.