On Questions on (Total) Domination Vertex Critical Graphs

Abstract

Let \(G = (V,E)\) be a graph. Let \(\gamma(G)\) and \(\gamma_t(G)\) be the domination and total domination number of a graph \(G\), respectively. The \(\gamma\)-criticality and \(\gamma_t\)-criticality of Harary graphs are studied. The Question \(2\) of the paper [W. Goddard et al., The Diameter of total domination vertex critical graphs, Discrete Math. \(286 (2004), 255-261]\) is fully answered with the family of Harary graphs. It is answered to the second part of Question \(1\) of that paper with some Harary graphs.