NP-hardness of multiple bondage in graphs

Let p≥2 be a positive integer. The p-bondage number of a graph G is the minimum number of edges whose removal from G results in a graph with larger p-domination number. The p-total bondage number of a graph G with no isolated vertex is the minimum number of non-pendant edges whose removal from G results in a graph with larger p-total domination number. The non-isolating p-bondage number of G is the minimum number of non-pendant edges whose removal from G results in a graph with larger p-domination number. In this paper we show that the decision problems for these multiple bondage problems are NP-hard.