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

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 $γ_t$ and maximum degree $Δ(G)$. Let $Δ= 3$ if $Δ(G) = 2$ and $Δ= Δ(G)$ if $Δ(G) \ge 3$. It is known [J. Graph Theory 49 (2005), 285--290; J. Graph Theory 54 (2007), 350--353] that $m \le Δ(n- γ_t)$. In this paper we characterize the extremal graphs $G$ satisfying $m = Δ(n- γ_t)$.

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