Matching and Factor-Critical Property in 3-Dominating-Critical Graphs

TL;DR

This paper investigates the matching properties of 3-dominating-critical graphs, establishing conditions under which such graphs have perfect or near-perfect matchings, thereby advancing understanding of their structural characteristics.

Contribution

It proves new theorems linking domination-criticality, graph order, and forbidden subgraphs to the existence of perfect or near-perfect matchings in these graphs.

Findings

Even order, $K_{1,6}$-free $ ightarrow$ perfect matching

Odd order, $K_{1,7}$-free $ ightarrow$ near perfect matching with three exceptions

Improves previous results on matching in domination-critical graphs

Abstract

Let $\gamma(G)$ be the domination number of a graph $G$. A graph $G$ is \emph{domination-vertex-critical}, or \emph{$\gamma$-vertex-critical}, if $\gamma(G-v)< \gamma(G)$ for every vertex $v \in V(G)$. In this paper, we show that: Let $G$ be a $\gamma$-vertex-critical graph and $\gamma(G)=3$. (1) If $G$ is of even order and $K_{1,6}$-free, then $G$ has a perfect matching; (2) If $G$ is of odd order and $K_{1,7}$-free, then $G$ has a near perfect matching with only three exceptions. All these results improve the known results.