Factor-Critical Property in 3-Dominating-Critical Graphs

Tao Wang 1 and Qinglin Yu 1, 2

1 Center for Combinatorics, Nankai University, Tianjin 300071, P. R. China
2 Department of Mathematics and Statistics, Thompson Rivers University, Kamloops, BC, Canada

Abstract     Full Text  PDF

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 numberof $G$ 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). This confirms a conjecture posed by Ananchuen and Plummer recently.