No 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].

A dominating set [Formula: see text] of a graph [Formula: see text] is said to be certified, if every vertex [Formula: see text] has either 0 neighbors or at least two neighbors in [Formula: see text]. The cardinality of a minimum certified dominating set of [Formula: see text] is called the certified domination number of [Formula: see text] and is denoted by [Formula: see text]. A graph [Formula: see text] is said to be [Formula: see text]-[Formula: see text] stable if for any two vertices [Formula: see text] such that [Formula: see text], we have [Formula: see text]. In this paper, we provide upper bounds for the [Formula: see text]-value of [Formula: see text]-[Formula: see text] stable graph. We also characterize the trees and unicyclic graphs that are [Formula: see text]-[Formula: see text] stable.

Let [Formula: see text] be a graph and [Formula: see text] be a Roman dominating function defined on [Formula: see text]. Let [Formula: see text] be some ordering of the vertices of [Formula: see text]. For any [Formula: see text], [Formula: see text] is defined by [Formula: see text]. If for all [Formula: see text], [Formula: see text], we have [Formula: see text], that is [Formula: see text], for some [Formula: see text], then [Formula: see text] is called a resolving Roman dominating function (RDF) on [Formula: see text]. The weight of a resolving RDF [Formula: see text] on [Formula: see text] is [Formula: see text]. The minimum weight of a resolving RDF on [Formula: see text] is called the resolving Roman domination number of [Formula: see text] and is denoted by [Formula: see text]. A resolving RDF on [Formula: see text] with weight [Formula: see text] is called a [Formula: see text]-function on [Formula: see text]. In this paper, we find the resolving Roman domination number of certain well-known classes of graphs. We also categorize the class of graphs whose resolving Roman domination number equals their order.

Let G = (V, E) be a graph and f : V → {0, 1, 2} be a function. The weight of a vertex u ∈ V is f (u) and a vertex u with weight f (u) = 0 is said to be undefended with respect to f , if it is not adjacent to a vertex with positive weight. The function f is a weak Roman dominating function (WRDF) if each vertex u with f (u) = 0 is adjacent to a vertex v with f (v) > 0 such that the functionThe weak Roman domination number, denoted by γ r (G), is the minimum weight of a weak Roman dominating function on G. In this paper we examine the effects on γ r (G) when G is modified by deleting a vertex. c