# Bounds on the Locating Roman Domination Number in Trees

• Volume: 38, Issue: 1, page 49-62
• ISSN: 2083-5892

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## Abstract

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A Roman dominating function (or just RDF) on a graph G = (V, E) is a function f : V → 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 weight of an RDF f is the value f(V (G)) = ∑u∈V(G) f(u). An RDF f can be represented as f = (V0, V1, V2), where Vi = v ∈ V : f(v) = i for i = 0, 1, 2. An RDF f = (V0, V1, V2) is called a locating Roman dominating function (or just LRDF) if N(u) ∩ V2 ≠ N(v) ∩ V2 for any pair u, v of distinct vertices of V0. The locating Roman domination number [...] γRL(G) ${\gamma }_{R}^{L}\left(G\right)$ is the minimum weight of an LRDF of G. In this paper, we study the locating Roman domination number in trees. We obtain lower and upper bounds for the locating Roman domination number of a tree in terms of its order and the number of leaves and support vertices, and characterize trees achieving equality for the bounds.

## How to cite

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Nader Jafari Rad, and Hadi Rahbani. "Bounds on the Locating Roman Domination Number in Trees." Discussiones Mathematicae Graph Theory 38.1 (2018): 49-62. <http://eudml.org/doc/288559>.

@article{NaderJafariRad2018,
abstract = {A Roman dominating function (or just RDF) on a graph G = (V, E) is a function f : V → 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 weight of an RDF f is the value f(V (G)) = ∑u∈V(G) f(u). An RDF f can be represented as f = (V0, V1, V2), where Vi = v ∈ V : f(v) = i for i = 0, 1, 2. An RDF f = (V0, V1, V2) is called a locating Roman dominating function (or just LRDF) if N(u) ∩ V2 ≠ N(v) ∩ V2 for any pair u, v of distinct vertices of V0. The locating Roman domination number [...] γRL(G) $\gamma _R^L (G)$ is the minimum weight of an LRDF of G. In this paper, we study the locating Roman domination number in trees. We obtain lower and upper bounds for the locating Roman domination number of a tree in terms of its order and the number of leaves and support vertices, and characterize trees achieving equality for the bounds.},
author = {Nader Jafari Rad, Hadi Rahbani},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {Roman domination number; locating domination number; locating Roman domination number; tree},
language = {eng},
number = {1},
pages = {49-62},
title = {Bounds on the Locating Roman Domination Number in Trees},
url = {http://eudml.org/doc/288559},
volume = {38},
year = {2018},
}

TY - JOUR
AU - Nader Jafari Rad
AU - Hadi Rahbani
TI - Bounds on the Locating Roman Domination Number in Trees
JO - Discussiones Mathematicae Graph Theory
PY - 2018
VL - 38
IS - 1
SP - 49
EP - 62
AB - A Roman dominating function (or just RDF) on a graph G = (V, E) is a function f : V → 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 weight of an RDF f is the value f(V (G)) = ∑u∈V(G) f(u). An RDF f can be represented as f = (V0, V1, V2), where Vi = v ∈ V : f(v) = i for i = 0, 1, 2. An RDF f = (V0, V1, V2) is called a locating Roman dominating function (or just LRDF) if N(u) ∩ V2 ≠ N(v) ∩ V2 for any pair u, v of distinct vertices of V0. The locating Roman domination number [...] γRL(G) $\gamma _R^L (G)$ is the minimum weight of an LRDF of G. In this paper, we study the locating Roman domination number in trees. We obtain lower and upper bounds for the locating Roman domination number of a tree in terms of its order and the number of leaves and support vertices, and characterize trees achieving equality for the bounds.
LA - eng
KW - Roman domination number; locating domination number; locating Roman domination number; tree
UR - http://eudml.org/doc/288559
ER -

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