Characterizations of Total Domination in R ( T )

A set [Formula: see text] of vertices is a total dominating set of a graph [Formula: see text] if every vertex of [Formula: see text] has at least one neighbor in [Formula: see text]. The total domination number [Formula: see text] is the minimum cardinality among all total dominating sets of [Formula: see text] The graph [Formula: see text] is obtained by adding a new vertex [Formula: see text] for each edge [Formula: see text] and joining [Formula: see text] to the end vertices [Formula: see text] and [Formula: see text] of [Formula: see text]. [S. Bermudo, Total domination on tree operators, Mediterr. J. Math. 20 (2023) 42.] proved that for every tree [Formula: see text] of order [Formula: see text] with [Formula: see text] leaves, [Formula: see text]. In this note, we provide a constructive characterization of extremal trees attaining these two bounds.