On Pairs of Disjoint Hop Dominating Sets in Graphs

A set S of vertices of a graph G is a hop dominating set of G if for every v V (G) S, v is at distance 2 from a vertex in S. The minimum cardinality ₕ (G) of a hop dominating set is the hop domination number of G. Any hop dominating set of cardinality ₕ (G) is a ₕ-set. A pair (S, T) of sets of vertices of G is a disjoint hop dominating pair if S T= and both S and T are hop dominating sets of G. In particular, if S is a ₕ-set, then T is an inverse hop dominating set of G. The minimum sum |S|+|T| among all disjoint hop dominating pairs is the disjoint hop domination number, denoted by ₇₇ (G). The minimum cardinality of an inverse hop dominating set of G is the inverse hop domination number of G, denoted by ₕ (G). In this paper, we initiate the study of inverse hop domination and disjoint hop domination. Interestingly, for every pair of positive integers m and n with 2 m n, there exists a connected graph G for which ₕ (G) =m and ₕ (G) =n. Also, for each positive integer n 4, there exists a connected graph G for which ₕ (G) +ₕ (G) -₇₇ (G) =n. Here we investigate these new concepts for some specific graphs including the join, corona and lexicographic product of graphs.