Transversal total hop domination in graphs

Let G be a finite simple graph. A set S subset of V (G) is a total hop dominating set of G if for every v is an element of V (G), there exists u is an element of S such that d(G)(u,v) = 2. Any total hop dominating set of G of minimum cardinality is a gamma(th)-set of G. A total hop dominating set S of G which intersects every gamma(th)-set of G is a transversal total hop dominating set. The minimum cardinality (gamma)over cap(th)(G) of a transversal total hop dominating set in G is the transversal total hop domination number of G. In this paper, we initiate the study of transversal total hop domination in graphs. First, we characterize a graph G of order n for which (gamma)over cap(th)(G) is n or n - 1, and also we determine the specific values of (gamma)over cap(th)(G) for some special graphs G. Next, we solve some realization problems involving (gamma)over cap(th)(G) with other parameters of G. Finally, we investigate the transversal total hop domination in the complementary prism, corona, and lexicographic product of graphs.