A constructive characterization of vertex cover Roman trees
Abel Cabrera Martínez, Dorota Kuziak, Ismael G. Yero · Discussiones Mathematicae Graph Theory · 2018
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 oiR (G) is the minimum weight w(f ) = vV (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 (G). A graph G is a vertex cover Roman graph