Abstract
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 γt 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−γt). In this paper we characterize the extremal graphs G satisfying m=Δ(n−γt).
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