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

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 $\Delta(G)\geq 9$.