A Constructive Characterization of Vertex Cover Roman Trees

Abstract

A Roman dominating function on a graph G = (V (G), E(G)) is a function f : V (G) → {0, 1, 2} satisfying the condition that every vertex u for which f (u) = 0 is adjacent to at least one vertex v for which f (v) = 2. The Roman dominating function f is an outer-independent Roman dominating function on G if the set of vertices labeled with zero under f is an independent set. The outer-independent Roman domination number γoiR(G) is the minimum weight w(f) = Σv∈V (G)f (v) of any outer-independent Roman dominating function f of G. A vertex cover of a graph G is a set of vertices that covers all the edges of G. The minimum cardinality of a vertex cover is denoted by α(G). A graph G is a vertex cover Roman graph if γoiR(G) = 2α(G). A constructive characterization of the vertex cover Roman trees is given in this article.

Keywords

• roman domination
• outer-independent roman domination
• vertex cover
• vertex independence
• trees
• 05c69
• 05c05
• 05c75