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.