An algorithm to recognize weak roman domination stable trees under vertex deletion

Abstract

Let [Formula: see text] be a graph and [Formula: see text] be a function. A vertex [Formula: see text] with weight [Formula: see text] is said to be undefended with respect to [Formula: see text], if it is not adjacent to any vertex with positive weight. The function [Formula: see text] is a weak Roman dominating function (WRDF) if each vertex [Formula: see text] with [Formula: see text] is adjacent to a vertex [Formula: see text] with [Formula: see text] such that the function [Formula: see text] defined by [Formula: see text], [Formula: see text] and [Formula: see text] if [Formula: see text], has no undefended vertex. The weight of [Formula: see text] is [Formula: see text]. The weak Roman domination number, denoted by [Formula: see text], is the minimum weight of a WRDF on [Formula: see text]. In this paper, we present two linear time algorithms one that obtains the weak Roman domination number of an arbitrary tree and the labeling of its vertices, to produce the weak Roman domination number, and the other, that determines whether the given tree is in [Formula: see text].