Varieties of Roman domination IV

Roman domination was introduced in 2004 by Cockayne, Dreyer, Hedetniemi, and Hedetniemi. If [Formula: see text] is the vertex set of a graph G, then a function [Formula: see text] is a Roman dominating function if every vertex [Formula: see text] for which [Formula: see text] is adjacent to at least one vertex v for which [Formula: see text]. The Roman domination number of a graph G, denoted by [Formula: see text], is the minimum weight of a Roman dominating function on G. In this work, we continue to survey what has been done on Roman domination. Mainly, we will present the signed version of some of these functions, and results on the Roman domination number when the structure of the graph is modified by addition of edges/vertices or removing edges/vertices, which are not contained in our previous four survey articles.