Equality in a Linear Vizing-Like Relation that Relates the Size and Total Domination Number of a Graph

Michael A. Henning, Ernst J. Joubert

Let $G$ be a graph each component of which has order at least 3, and let $G$ have order $n$, size $m$, total domination number ${\gamma }_t$ and maximum degree $\Delta (G)$. Let $\Delta =3$ if $Δ(G)…