Weak Roman subdivision number of graphs

Abstract

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.