Varieties of Roman domination II

In this work, we continue to survey what has been done on the Roman domination. More precisely, we will present in two sections several variations of Roman dominating functions as well as the signed version of some of these functions. It should be noted that a first part of this survey comprising 9 varieties is published as a chapter book in “Topics in domination in graphs” edited by T.W. Haynes, S.T. Hedetniemi and M.A. Henning. We recall that a function (Formula presented.) is a Roman dominating function (or just RDF) if every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. The Roman domination number of a graph G, denoted by (Formula presented.) is the minimum weight of an RDF on G.