Total $k$-Uniform Graphs

Abstract

A sequence of vertices in a graph $G$ without isolated vertices is called a total dominating sequence if every vertex $v$ in the sequence has a neighbor which is adjacent to no vertex preceding $v$ in the sequence, and at the end every vertex of $G$ has at least one neighbor in the sequence. Minimum and maximum lengths of a total dominating sequence is the total domination number of $G$ (denoted by $\gamma_t(G)$) and the Grundy total domination number of $G$ (denoted by $\gamma_{gr}^t(G)$), respectively. In this paper, we study graphs with equal total domination number and Grundy total domination number. For every positive integer $k$, we call $G$ a total $k$-uniform graph if $\gamma_t(G)=\gamma_{gr}^t(G)=k$. We prove that there is no total $k$-uniform graph when $k$ is odd. In addition, we present a total 4-uniform graph which stands as a counterexample for a conjecture by T. Gologranc et.al. and provide a connected total 8-uniform graph. Moreover, we prove that every total $k$-uniform, connected and false twin-free graph is regular for every even $k$. We also show that there is no total $k$-uniform chordal connected graph with $k\geq 4$ and characterize all total $k$-uniform chordal graphs.