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.