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)≥3. It is known [M.A. Henning, A linear Vizing-like relation relating the size and total domination number of a graph, J. Graph Theory 49 (2005) 285-290; E. Shan, L. Kang, M.A. Henning, Erratum to: a linear Vizing-like relation relating the size and total domination number of a graph, J. Graph Theory 54 (2007) 350-353] that m≤Δ(n-γ_t). In this paper we characterize the extremal graphs G satisfying m=Δ(n- γ_t).