On the complexity of the bondage and reinforcement problems

Abstract

Let $G=(V,E)$ be a graph. A subset $D\subseteq V$ 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.