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

Abstract

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 $$k\ge 1$$ be an integer, and a function $$f:V \rightarrow \{0,1,\dots ,k\}$$ defined on V called a Roman $$\{k\}$$ -dominating function if for every vertex $$v\in V$$ with $$f(v)=0$$ , $$\sum _{u\in N(v)}f(u)\ge k$$ , where N(v) is the open neighborhood of v in G. The minimum value $$\sum _{u\in V}f(u)$$ for a Roman $$\{k\}$$ -dominating function f on G is called the Roman $$\{k\}$$ -domination number of G, denoted by $$\gamma _{\{Rk\}}(G)$$ . We first present bounds on $$\gamma _{\{Rk\}}(G)$$ in terms of other domination parameters, including $$\gamma _{\{Rk\}}(G)\le k\gamma (G)$$ . 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 $$k\in \mathbb {Z_{+}}$$ , associated decision problem for the Roman $$\{k\}$$ -domination is NP-complete, even for bipartite planar graphs, chordal bipartite graphs and undirected path graphs.