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 γₕ(G) of a hop dominating set is the hop domination number of G. Any hop dominating set of cardinality γₕ(G) is a γₕ-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 γₕ-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 γₕₕ(G). The minimum cardinality of an inverse hop dominating set of G is the inverse hop domination number of G, denoted by γₕ(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 γₕ(G)=m and γₕ(G)=n. Also, for each positive integer n≥ 4, there exists a connected graph G for which γₕ(G)+γₕ(G)-γₕₕ(G)=n. Here we investigate these new concepts for some specific graphs including the join, corona and lexicographic product of graphs.