Roman Domination Number of Split Graphs of Certain Graphs

Abstract

A dominating set D in a graph G = (V;E) is a subset of vertices such that every vertex not in D is adjacent to at least one vertex in D. The minimum cardinality of such a set is called the dominating number of G. A Roman dominating function on G is a function f : V ! f0;1;2g with the property that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. The weight of a Roman dominating function f is defined as w(f) = ∑ v2V f(v); and the minimum possible weight over all Roman dominating functions on G is called the Roman dominating number, denoted by gR(G). We also study the split graph SG of G, constructed by adding, for each vertex v 2V(G), a new vertex v0 and making v0 adjacent to every neighbor of v. In this work, we explore how different graphs relate to their corresponding split graphs SG. For convenience, we label the original vertices of G by A;B;:::;Z and the new vertices of SG by A0;B0;:::;Z0. Our main focus is on the Roman domination number of split graphs derived from the following well-known families of graphs: path graphs Pn, cycle graphs Cn, complete graphs Kn, wheel graphs Wn, double-wheel graphs DWn, fan graphs Fn, star graphs Sn, circulant graphs Cnh1;2i, Cnh1;3i, Cnh1;4i, Cnh1;2;3i, prism graphs Pn, anti-prism graphs An, and complete bipartite graphs K2;m.