Roman {k}-domination in trees and complexity results for some classes of graphs

In this paper, we study Roman { k }-dominating functions on a graph G with vertex set V for a positive integer k : a variant of { k }-dominating functions, generations of Roman { 2 } -dominating functions and the characteristic functions of dominating sets, respectively, which unify classic domination parameters with certain Roman domination parameters on G . Let k ≥ 1 be an integer, and a function f : V → { 0 , 1 , ⋯ , k } defined on V called a Roman { k } -dominating function if for every vertex v ∈ V with f ( v ) = 0 , ∑ u ∈ N ( v ) f ( u ) ≥ k , where N ( v ) is the open neighborhood of v in G . The minimum value ∑ u ∈ V f ( u ) for a Roman { k } -dominating function f on G is called the Roman { k } -domination number of G , denoted by γ { R k } ( G ) . We first present bounds on γ { R k } ( G ) in terms of other domination parameters, including γ { R k } ( G ) ≤ k γ ( G ) . Secondly, we show one of our main results: characterizing the trees achieving equality in the bound mentioned above, which generalizes M.A. Henning and W.F. klostermeyer’s results on the Roman {2}-domination number (Henning and Klostermeyer in Discrete Appl Math 217:557–564, 2017). Finally, we show that for every fixed k ∈ Z + , associated decision problem for the Roman { k } -domination is NP-complete, even for bipartite planar graphs, chordal bipartite graphs and undirected path graphs.