Roman Domination in the Shadow Distance Graphs

Abstract

A function ψ : V → {0, 1, 2} satisfying the requirement that each vertex x for which ψ(x) = 0 is adjacent to at least one vertex y for which ψ( y) = 2 is known as a Roman dominating function (Rdf) on a graph. A Rdf's weight is represented by the value ψ( y) = x∈V ψ(x). The Roman domination number (Rdn) of a graph G is the minimal weight of a Rdf on that graph. In this article, we establish Rdn for the shadow distance graph of the path, cycle, and star graphs with predetermined distance sets.