On the complexity of the bondage and reinforcement problems

Let G=(V,E) be a graph. A subset D@?V is a dominating set if every vertex not in D is adjacent to a vertex in D. A dominating set D is called a total dominating set if every vertex in D is adjacent to a vertex in D. The domination (resp. total domination) number of G is the smallest cardinality of a dominating (resp. total dominating) set of G. The bondage (resp. total bondage) number of a nonempty graph G is the smallest number of edges whose removal from G results in a graph with larger domination (resp. total domination) number of G. The reinforcement (resp. total reinforcement) number of G is the smallest number of edges whose addition to G results in a graph with smaller domination (resp. total domination) number. This paper shows that the decision problems for the bondage, total bondage, reinforcement and total reinforcement numbers are all NP-hard.