On perfect Roman domination number in trees: complexity and bounds

Abstract

A perfect Roman dominating function on a graph $$G =(V,E)$$ is a function $$f: V \longrightarrow \{0, 1, 2\}$$ satisfying the condition that every vertex u with $$f(u) = 0$$ is adjacent to exactly one vertex v for which $$f(v)=2$$ . The weight of a perfect Roman dominating function f is the sum of the weights of the vertices. The perfect Roman domination number of G, denoted by $$\gamma _{R}^{p}(G)$$ , is the minimum weight of a perfect Roman dominating function in G. In this paper, we first show that the decision problem associated with $$\gamma _{R}^{p}(G)$$ is NP-complete for bipartite graphs. Then, we prove that for every tree T of order $$n\ge 3$$ , with $$\ell $$ leaves and s support vertices, $$\gamma _R^P(T)\le (4n-l+2s-2)/5$$ , improving a previous bound given in Henning et al. (Discrete Appl Math 236:235–245, 2018).