Let be a graph. A subset is a dominating set if every vertex not in is adjacent to a vertex in . A dominating set is called a total dominating set if every vertex in is adjacent to a vertex in . The domination (resp. total domination) number of is the smallest cardinality of a dominating (resp. total dominating) set of . The bondage (resp. total bondage) number of a nonempty graph is the smallest number of edges whose removal from results in a graph with larger domination (resp. total domination) number of . The reinforcement (resp. total reinforcement) number of is the smallest number of edges whose addition to 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.
Highlights
► The bondage for measuring the vulnerability of the network domination under link failure. ► The reinforcement for measuring the stability of the network domination under link addition. ► We show that the decision problems for the bondage and the reinforcement are NP-hard.
The work was supported by NNSF of China (No. 11071233), and reported in International Conference on Graph Theory, Combinatorics and Applications, October 29–November 2, 2010, Jinhua Zhejiang, China.