On The Bondage Number of Middle Graphs

Let $G = (V(G), E(G))$ be a simple graph. A subset $S$ of $V(G)$ is a dominating set of $G$ if, for any vertex $v \in {V(G)-S}$, there exists some vertex $u \in S$ such that $uv \in E(G)$. The domination number, denoted by $\gamma(G)$, is the cardinality of a minimal dominating set of $G$. There are several types of domination parameters depending upon the nature of domination and the nature of dominating set. These parameters are bondage, reinforcement, strong-weak domination, strong-weak bondage numbers. In this paper, we first investigate the strong-weak domination number of middle graphs of a graph. Then several results for the bondage, strong-weak bondage number of middle graphs are obtained.