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 \(\gamma_t(G) \leq \frac{3}{5}n(G)\) for graphs with unique \(\gamma_t\)-sets, and we characterize all graphs with unique \(\gamma_t\)-sets where \(\gamma_t(G) = \frac{3}{5}n(G)\).