Factor-Critical Property in 3-Dominating-Critical Graphs

Abstract

A vertex subset $S$ of a graph $G$ is a dominating set if every vertex of $G$ either belongs to $S$ or is adjacent to a vertex of $S$. The cardinality of a smallest dominating set is called the dominating number of $G$ and is denoted by $\gamma (G)$. A graph $G$ is said to be $\gamma$- vertex-critical if $\gamma (G-v)<\gamma (G)$, for every vertex $v$ in $G$. Let $G$ be a 2-connected $K_{1,5}$-free 3-vertex-critical graph. For any vertex $v\in V(G)$, we show that $G-v$ has a perfect matching (except two graphs), which is a conjecture posed by Ananchuen and Plummer.

8 pages

Cited by in corpus (1)

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