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).