Transversal total hop domination in graphs

Abstract

Let [Formula: see text] be a finite simple graph. A set [Formula: see text] is a total hop dominating set of [Formula: see text] if for every [Formula: see text], there exists [Formula: see text] such that [Formula: see text]. Any total hop dominating set of [Formula: see text] of minimum cardinality is a [Formula: see text]-set of [Formula: see text]. A total hop dominating set [Formula: see text] of [Formula: see text] which intersects every [Formula: see text]-set of [Formula: see text] is a transversal total hop dominating set. The minimum cardinality [Formula: see text] of a transversal total hop dominating set in [Formula: see text] is the transversal total hop domination number of [Formula: see text]. In this paper, we initiate the study of transversal total hop domination in graphs. First, we characterize a graph [Formula: see text] of order [Formula: see text] for which [Formula: see text] is [Formula: see text] or [Formula: see text], and also we determine the specific values of [Formula: see text] for some special graphs [Formula: see text]. Next, we solve some realization problems involving [Formula: see text] with other parameters of [Formula: see text]. Finally, we investigate the transversal total hop domination in the complementary prism, corona, and lexicographic product of graphs.