On a Class of Graphs with Large Total Domination Number

Let $\gamma (G)$ and ${\gamma }_t(G)$ denote the domination number and the total domination number, respectively, of a graph $G$ with no isolated vertices. It is well-known that ${\gamma }_t(G)\le 2\gamma (G)$. We provide a characterization of a large family of graphs (including chordal graphs) satisfying $\gamma_t(G)= 2\gamma(G)$, strictly generalizing the results of Henning (2001) and Hou et al. (2010), and partially answering an open question of Henning (2009).