On the 3-$\gamma_t$-Critical Graphs of Order $\Delta(G)+3$

Haoli Wang; Xirong Xu; Yang Yuansheng; Lei Wang

On the 3-$\gamma_t$-Critical Graphs of Order $\Delta(G)+3$

Combinatorics 2011-03-15 v1

Authors: Haoli Wang , Xirong Xu , Yang Yuansheng , Lei Wang

Abstract

Let $\gamma_t(G)$ be the total domination number of graph $G$ , a graph $G$ is $k$ -total domination vertex critical (or\ just\ $k$ - $\gamma_t$ -critical) if $\gamma_t(G)=k$ , and for any vertex $v$ of $G$ that is not adjacent to a vertex of degree one, $\gamma_t(G-v)=k-1$ . Mojdeh and Rad \cite{MR06} proposed an open problem: Does there exist a 3- $\gamma_t$ -critical graph $G$ of order $\Delta(G)+3$ with $\Delta(G)$ odd? In this paper, we prove that there exists a 3- $\gamma_t$ -critical graph $G$ of order $\Delta(G)+3$ with odd $\Delta(G)\geq 9$ .

Keywords

total domination number 3-$\gamma_t$-critical graph maximum degree vertex critical

Cite

@article{arxiv.1103.2415,
  title  = {On the 3-$\gamma_t$-Critical Graphs of Order $\Delta(G)+3$},
  author = {Haoli Wang and Xirong Xu and Yang Yuansheng and Lei Wang},
  journal= {arXiv preprint arXiv:1103.2415},
  year   = {2011}
}

Comments

This paper was accpted by Utilitas Mathematica in 2008

On the 3-$\gamma_t$-Critical Graphs of Order $\Delta(G)+3$

Abstract

Keywords

Cite

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