Upper Bound on the Diameter of a Total Domination Vertex Critical Graph

Abstract

A vertex of a graph is said to be total domination critical if its deletion decreases the total domination number. A graph is said to be total domination vertex critical if all of its vertices, except the supporting vertices, are total domination vertex critical. We show that if \(G\) is a connected total domination vertex critical graph with total domination number \(k \geq 4\), then the diameter of \(G\) is at most \(\lfloor \frac{5k-7}{3}\rfloor\).