(untitled) (European Journal of Pure and Applied Mathematics, 2025)
A set S of vertices of a graph G is a hop dominating set of G if for every vV(G)S, v is at distance 2 from a vertex in S. The minimum cardinality γ_h(G) of a hop dominating set is the hop domination number of G. Any hop dominating set of cardinality γ_h(G) is a γ_h-set. A pair (S,T) of sets of vertices of G is a disjoint hop dominating pair if ST= and both S and T are hop dominating sets of G. In particular, if S is a γ_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 γ_hh(G). The minimum cardinality of an inverse hop dominating set of G is the inverse hop domination number of G, denoted by γ_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 2mn, there exists a connected graph G for which γ_h(G)=m and γ_h(G)=n. Also, for each positive integer n4, there exists a connected graph G for which γ_h(G)+γ_h(G)-γ_hh(G)=n. Here we investigate these new concepts for some specific graphs including the join, corona and lexicographic product of graphs.