NP-completeness of some generalized hop and step domination parameters in graphs

Let r ≥ 2. A subset S of vertices of a graph G is a r-hop independent dominating set if every vertex outside S is at distance r from a vertex of S, and for any pair v, w ∈ S, d(v, w) ≠ r. A r-hop Roman dominating function (rHRDF) is a function f on V (G) with values 0, 1 and 2 having the property that for every vertex v ∈ V with f(v) = 0 there is a vertex u with f(u) = 2 and d(u, v) = r. A r-step Roman dominating function (rSRDF) is a function f on V (G) with values 0, 1 and 2 having the property that for every vertex v with f(v) = 0 or 2, there is a vertex u with f(u) = 2 and d(u, v) = r. A rHRDF f is a r-hop Roman independent dominating function if for any pair v, w with non-zero labels under f, d(v, w) ≠ r. We show that the decision problem associated with each of r-hop independent domination, r-hop Roman domination, r-hop Roman independent domination and r-step Roman domination is NP-complete even when restricted to planar bipartite graphs or planar chordal graphs.