Factor-Critical Property in 3-Dominating-Critical Graphs

Details Factor-Critical Property in 3-Dominating-Critical Graphs Factor-Critical Property in 3-Dominating-Critical Graphs

2006-08-28

arxiv.org/abs/math/0608672v1

Mathematics

Combinatorics

8 pages

Scientific paper

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.

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