Characterization of perfect Roman domination edge critical trees
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 study the
graphs for which adding any new edge decreases the perfect Roman
domination number. We call these graphs $\gamma_R^p$-edge critical.
The purpose of this paper is to characterize the class of $\gamma_R^p$-edge critical trees.
Copyright (c) 2019 Abdollah Alhevaz, Mahsa Darkooti, Sadegh Rahimi, Hadi Rahbani
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