Total Domination Edge Critical Graphs with Minimum Diameter

Abstract

Denote the total domination number of a graph \(G\) by \(\gamma_t(G)\). A graph \(G\) is said to be total domination edge critical, or simply \(\gamma_t\)-critical, if \(\gamma_t(G+e) < \gamma_t(G)\) for each edge \(e \in E(\overline{G})\). For \(\gamma_t\)-critical graphs \(G\), that is, \(\gamma_t\)-critical graphs with \(\gamma_t(G) = 3\), the diameter of \(G\) is either \(2\) or \(3\). We study the \(3_t\)-critical graphs \(G\) with \(diam(G) = 2\).