\(4_{t}\)-Critical Graphs with Maximum Diameter

Let \( \gamma_t(G) \) denote the total domination number of the graph \( G \). 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 \( e \in E(\overline{G}) \). We show that, for \( 4_t \)-critical graphs \( G \), that is, \( \gamma_t \)-critical graphs with \( \gamma_t(G) = 4 \), the diameter of \( G \) is either \( 2 \), \( 3 \), or \( 4 \). Further, we characterize structurally the \( 4_t \)-critical graphs \( G \) with \( \text{diam}(G) = 4 \).