Roman Labeling in Digraphs

Abstract

Let G be a connected graph. Roman labeling of G is a function $f: V(G) \to \{0, 1, 2\}$ such that any vertex with label 0 is adjacent to a vertex with label 2. The function value $f(v)$ of a vertex v of the graph G is called the label of v. Weight of a Roman labeling, f is defined as the sum of all vertex labels.That is, $\mathrm{w}\left(f\right)=\sum\limits_{v\in V(G)}f(v)$. Roman number of a graph G is defined as the minimum weight of a Roman labeling on G and is denoted by $S(G)$. It was introduced by Satheesh and Suresh Kumar [3]. In this paper, we extend the idea of Roman labeling and Roman number to Digraphs.