Transversal Hop Domination in Graphs

Let G be a graph. A set S C V (G) is a hop dominating set of G if for every v E V (G)\S, there exists u E S such that dG(u, v) = 2. The minimum cardinality gamma h(G) of a hop dominating set is the hop domination number of G. Any hop dominating set of G of cardinality gamma h(G) is a gamma h-set of G. A hop dominating set S of G which intersects every gamma h-set of G is a transversal hop dominating set. The minimum cardinality gamma bh(G) of a transversal hop dominating set in G is the transversal hop domination number of G. In this paper, we initiate the study of transversal hop domination. First, we characterize graphs G whose values for gamma bh(G) are either n or n - 1, and we determine the specific values of gamma bh(G) for some specific graphs. Next, we show that for every positive integers a and b with a > 2 and b > 3a, there exists a connected graph G on b vertices such that gamma bh(G) = a. We also show that for every positive integers a and b with 2 < a < b, there exists a connected graph G for which gamma h(G) = a and gamma bh(G) = b. Finally, we investigate the transversal hop dominating sets in the join and corona of two graphs, and determine their corresponding transversal hop domination numbers.