Equality in a linear Vizing-like relation that relates the size and total domination number of a graph

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 ∆(G). Let ∆ = 3 if ∆(G) = 2 and ∆ = ∆(G) if ∆(G) ≥ 3. It is known [J. Graph Theory 49 (2005), 285-290; J. Graph Theory 54 (2007), 350-353] that m ≤ ∆(n − γt). In this paper we characterize the extremal graphs G satisfying m = ∆(n − γt).

Optimal linear‐Vizing relationships for (total) domination in graphs

“…The connected graphs that achieve equality in the upper bound of Theorem 9 were characterized in 2013 by Henning and Joubert [13]. As a consequence of their characterization, if the bounded maximum degree $${\rm{\Delta }}\ge 4$$, then we have strict inequality in the upper bound of Theorem 9.…”