%0 Journal Article
%T Roman domination excellent graphs: trees
%J Communications in Combinatorics and Optimization
%I Azarbaijan Shahid Madani University
%Z 2538-2128
%A Samodivkin, Vladimir D.
%D 2018
%\ 06/01/2018
%V 3
%N 1
%P 1-24
%! Roman domination excellent graphs: trees
%K Roman domination number
%K excellent graphs
%K trees
%R 10.22049/cco.2017.25806.1041
%X A Roman dominating function (RDF) on a graph $G = (V, E)$ is a labeling $f : V rightarrow {0, 1, 2}$ suchthat every vertex with label $0$ has a neighbor with label $2$. The weight of $f$ is the value $f(V) = Sigma_{vin V} f(v)$The Roman domination number, $gamma_R(G)$, of $G$ is theminimum weight of an RDF on $G$.An RDF of minimum weight is called a $gamma_R$-function.A graph G is said to be $gamma_R$-excellent if for each vertex $x in V$there is a $gamma_R$-function $h_x$ on $G$ with $h_x(x) not = 0$. We present a constructive characterization of $gamma_R$-excellent trees using labelings. A graph $G$ is said to be in class $UVR$ if $gamma(G-v) = gamma (G)$ for each $v in V$, where $gamma(G)$ is the domination number of $G$. We show that each tree in $UVR$ is $gamma_R$-excellent.
%U http://comb-opt.azaruniv.ac.ir/article_13654_ec8df599ae3874367c247f6fb520698c.pdf