Geodesic
A Constructive Characterization of Vertex Cover Roman Trees
Discussiones Mathematicae. Graph Theory, Tome 41 (2021) no. 1, pp. 267-283

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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 
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Martínez, Abel Cabrera; Kuziak, Dorota; Yero, Ismael G. A Constructive Characterization of Vertex Cover Roman Trees. Discussiones Mathematicae. Graph Theory, Tome 41 (2021) no. 1, pp. 267-283. http://geodesic.mathdoc.fr/item/DMGT_2021_41_1_a16/