Effect of vertex deletion on the weak Roman domination number of a graph

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].

show abstract

Weak Roman subdivision number of graphs

Pushpam

Srilakshmi

2021

Discrete Math. Algorithm. Appl.

View full text Add to dashboard Cite

A Roman dominating function (RDF) on a graph [Formula: see text] is a function [Formula: see text] satisfying the condition that every vertex u for which [Formula: see text] is adjacent to at least one vertex v for which [Formula: see text]. A vertex [Formula: see text] with [Formula: see text] is undefended if it is not adjacent to a vertex with [Formula: see text]. 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 Roman domination subdivision number of a graph [Formula: see text] is the minimum number of edges that must be subdivided in order to increase the Roman domination number of [Formula: see text]. We introduce the concept of weak Roman subdivision number of a graph [Formula: see text], denoted by [Formula: see text] as the minimum number of edges that must be subdivided in order to increase the weak Roman domination number of [Formula: see text]. In this paper, we determine the exact values of the weak Roman subdivision number for paths, cycles and complete bipartite graphs. We obtain bounds for the weak Roman subdivision number of a graph and characterize the extremal graphs.

show abstract

Effect of vertex deletion on the weak Roman domination number of a graph

Cited by 4 publications

References 9 publications

Weak Roman Bondage Number of a Graph

Weak Roman Bondage Number of a Graph

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

Weak Roman subdivision number of graphs

Contact Info

Product

Resources

About