Calculating Modern Roman Domination of Fan Graph and Double Fan Graph

Abstract

This paper is concerned with the concept of modern Roman domination in graphs. A Modern Roman dominating function on a graph is labeling such that every vertex with label 0 is adjacent to two vertices; one of them of label 2 and the other of label 3 and every vertex with label 1 is adjacent to a vertex with label 2 or label 3. The weight of a Roman dominating function is the value 𝑓(𝑉) = ∑ 𝑓(𝑣) 𝑣∈𝑉. The minimum weight of all possible Roman dominating functions is called the "Roman Domination Number" of a graph. This dominance can be used in many aspects of life, for example in computer networks, transmission lines, and many others. In this paper, the modern Roman domination of the fan graph and the double fan graph with their complement are determined. Also, it has been determined the the number of modern Roman dominations of the corona of two specific graphs like the corone of two fan graph, two double fan graph ,fan graph and double fan graph and the oppisit of them.