(untitled) (European Journal of Pure and Applied Mathematics, 2025)
Let G be an undirected connected graph with vertex and edge sets V(G) and E(G), respectively. A hop dominating set S in G is 2-step movable hop dominating if for each v S, S{v} or [S{v}] {w} for some w [V(G)S] N_G^2(v) is a hop dominating set in G. The minimum cardinality of a 2-step movable hop dominating set in G, denoted by γ_mh^2(G), is called the 2-step movable hop domination number of G. In this paper, we characterize those graphs which admit a 2-step movable hop dominating set. We give bounds on the 2-step movable hop domination number and give necessary and sufficient conditions for those graphs that attain these bounds. We also characterize the 2-step movable hop dominating sets in the shadow graph and complementary prism and determine their respective 2-step movable hop domination number.