Graphs and Their Complements With Equal Total Domination Numbers

Abstract

A set $S$ of vertices in a graph $G$ is a total dominating set of $G$ if every vertex of $G$ is adjacent to some vertex in $S$. The minimum cardinality of a total dominating set of $G$ is the total domination number of $G$. We study graphs having the same total domination number as their complements. In particular, we characterize the cubic graphs having this property. Also, we characterize such graphs with total domination numbers equal to two or three, and we determine properties of the ones with larger total domination numbers.