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Communications in Combinatorics and Optimization
Articles in Press
Current Issue
Journal Archive
Volume Volume 3 (2018)
Issue Issue 1
Volume Volume 2 (2017)
Volume Volume 1 (2016)
Samodivkin, V. (2018). Roman domination excellent graphs: trees. Communications in Combinatorics and Optimization, 3(1), 1-24. doi: 10.22049/cco.2017.25806.1041
Vladimir D. Samodivkin. "Roman domination excellent graphs: trees". Communications in Combinatorics and Optimization, 3, 1, 2018, 1-24. doi: 10.22049/cco.2017.25806.1041
Samodivkin, V. (2018). 'Roman domination excellent graphs: trees', Communications in Combinatorics and Optimization, 3(1), pp. 1-24. doi: 10.22049/cco.2017.25806.1041
Samodivkin, V. Roman domination excellent graphs: trees. Communications in Combinatorics and Optimization, 2018; 3(1): 1-24. doi: 10.22049/cco.2017.25806.1041

Roman domination excellent graphs: trees

Article 1, Volume 3, Issue 1, Winter and Spring 2018, Page 1-24  XML PDF (617 K)
Document Type: Original paper
DOI: 10.22049/cco.2017.25806.1041
Author
Vladimir D. Samodivkin
University of Architecture, Civil Еngineering and Geodesy; Department of Mathematics
Abstract
A Roman dominating function (RDF) on a graph $G = (V, E)$
is a labeling $f : V rightarrow {0, 1, 2}$ such
that 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 the
minimum 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.
Keywords
Roman domination number; excellent graphs; trees
Main Subjects
Graph theory
Statistics
Article View: 262
PDF Download: 164
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