Partial domination of middle graphs

Abstract

For any graph $G=(V,E)$, a subset $S\subseteq V$ is called {\it an isolating set} of $G$ if $V\setminus N_G[S]$ is an independent set of $G$, where $N_G[S]=S\cup N_G(S)$, and {\it the isolation number} of $G$, denoted by ${\hat{I}}^1(G)$, is the size of a smallest isolating set of $G$. In this article, we show that the isolation number of the middle graph of $G$ is equal to the size of a smallest maximal matching of $G$.

16 Pages and 6 figures