Roman Domination in the Shadow Distance Graphs - Kallesh, K.; Kumar, U. Vijaya Chandra; Murali, R. (Communications in Mathematics and Applications, 2023)
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 \(ψ(y)= _xV ψ(x)\). 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.