On the complexity of some hop domination parameters

A hop Roman dominating function (HRDF) on a graph G = (V, E) is a function f : V -> {0, 1, 2} having the property that for every vertex v is an element of V with f(v) = 0 there is a vertex u with f(u) = 2 and d(u, v) = 2. The weight of an HRDF f is the sum of its values on V. The minimum weight of an HRDF on G is called the hop Roman domination number of G. An HRDF f is a hop Roman independent dominating function (HRIDF) if for any pair v, w with f(v) > 0 and f(w) > 0, d(v, w) not equal 2. The minimum weight of an HRIDF on G is called the hop Roman independent domination number of G. In this paper, we study the complexity of the hop independent dominating problem, the hop Roman domination function problem and the hop Roman independent domination function problem, and show that the decision problem for each of the above three problems is NP-complete even when restricted to planar bipartite graphs or planar chordal graphs.