Partial domination of middle graphs

TL;DR

This paper establishes a precise relationship between the isolation number of a graph's middle graph and the size of its smallest maximal matching, providing new insights into graph domination concepts.

Contribution

It proves that the isolation number of the middle graph equals the size of the smallest maximal matching in the original graph, linking two graph parameters.

Findings

Isolation number of middle graph equals smallest maximal matching size

Provides a new characterization of middle graph domination

Connects domination concepts with matching theory

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 $\iota(G)$, 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 $G$.