Transversal Hop Domination in Graphs

Abstract

Let G be a graph. A set S ⊆ V (G) is a hop dominating set of G if for every v ∈ V (G)\S, there exists u ∈ S such that dG(u, v) = 2. The minimum cardinality γh(G) of a hop dominating set is the hop domination number of G. Any hop dominating set of G of cardinality γh(G) is a γh-set of G. A hop dominating set S of G which intersects every γh-set of G is a transversal hop dominating set. The minimum cardinality γbh(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 γbh(G) are either n or n − 1, and we determine the specific values of γbh(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 γbh(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 γh(G) = a and γbh(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.