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.