Unique Total Domination Graphs

Abstract

A set $D$ of vertices in a graph $G$ is a total dominating set if every vertex of $G$ has at least one neighbor in $D$. The minimum cardinality of a total dominating set of $G$ is called the total domination number of $G$, denoted by ${\gamma }_t(G)$. A total dominating set of $G$ with cardinality ${\gamma }_t(G)$ is called a ${\gamma }_t$-set of $G$. We characterize trees with unique ${\gamma }_t$-sets. Further, we prove that ${\gamma }_t(G)\le \frac{3}{5}n(G)$ for graphs with unique ${\gamma }_t$-sets, and we characterize all graphs with unique ${\gamma }_t$-sets where ${\gamma }_t(G)=\frac{3}{5}n(G)$.