Bounding the rainbow domination number of a tree in terms of its annihilation number

A {em 2-rainbow dominating function} (2RDF) of a graph $G$ is a function $f$ from the vertex set $V(G)$ to the set of all subsets of the set ${1,2}$ such that for any vertex $vin V(G)$ with $f(v)=emptyset$ the condition $bigcup_{uin N(v)}f(u)={1,2}$ is fulfilled, where $N(v)$ is the open neighborhoo...

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Main Authors: Nasrin Dehgardi, Mahmoud Sheikholeslami, Abdollah Khodkar
Format: Article
Language:English
Published: University of Isfahan 2013-09-01
Series:Transactions on Combinatorics
Subjects:
Online Access:http://www.combinatorics.ir/?_action=showPDF&article=3051&_ob=dc39b3b99937a3eea4c41cc51272e53a&fileName=full_text.pdf.
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spelling doaj-c1d862b6e5de48e6bbef8eba63f6f5ea2020-11-25T00:05:23ZengUniversity of IsfahanTransactions on Combinatorics2251-86572251-86652013-09-01232132Bounding the rainbow domination number of a tree in terms of its annihilation numberNasrin DehgardiMahmoud SheikholeslamiAbdollah KhodkarA {em 2-rainbow dominating function} (2RDF) of a graph $G$ is a function $f$ from the vertex set $V(G)$ to the set of all subsets of the set ${1,2}$ such that for any vertex $vin V(G)$ with $f(v)=emptyset$ the condition $bigcup_{uin N(v)}f(u)={1,2}$ is fulfilled, where $N(v)$ is the open neighborhood of $v$. The {em weight} of a 2RDF $f$ is the value $omega(f)=sum_{vin V}|f (v)|$. The {em $2$-rainbow domination number} of a graph $G$, denoted by $gamma_{r2}(G)$, is the minimum weight of a 2RDF of G. The {em annihilation number} $a(G)$ is the largest integer $k$ such that the sum of the first $k$ terms of the non-decreasing degree sequence of $G$ is at most the number of edges in $G$. In this paper, we prove that for any tree $T$ with at least two vertices, $gamma_{r2}(T)le a(T)+1$.http://www.combinatorics.ir/?_action=showPDF&article=3051&_ob=dc39b3b99937a3eea4c41cc51272e53a&fileName=full_text.pdf.annihilation number2-rainbow dominating function2-rainbow domination number
collection DOAJ
language English
format Article
sources DOAJ
author Nasrin Dehgardi
Mahmoud Sheikholeslami
Abdollah Khodkar
spellingShingle Nasrin Dehgardi
Mahmoud Sheikholeslami
Abdollah Khodkar
Bounding the rainbow domination number of a tree in terms of its annihilation number
Transactions on Combinatorics
annihilation number
2-rainbow dominating function
2-rainbow domination number
author_facet Nasrin Dehgardi
Mahmoud Sheikholeslami
Abdollah Khodkar
author_sort Nasrin Dehgardi
title Bounding the rainbow domination number of a tree in terms of its annihilation number
title_short Bounding the rainbow domination number of a tree in terms of its annihilation number
title_full Bounding the rainbow domination number of a tree in terms of its annihilation number
title_fullStr Bounding the rainbow domination number of a tree in terms of its annihilation number
title_full_unstemmed Bounding the rainbow domination number of a tree in terms of its annihilation number
title_sort bounding the rainbow domination number of a tree in terms of its annihilation number
publisher University of Isfahan
series Transactions on Combinatorics
issn 2251-8657
2251-8665
publishDate 2013-09-01
description A {em 2-rainbow dominating function} (2RDF) of a graph $G$ is a function $f$ from the vertex set $V(G)$ to the set of all subsets of the set ${1,2}$ such that for any vertex $vin V(G)$ with $f(v)=emptyset$ the condition $bigcup_{uin N(v)}f(u)={1,2}$ is fulfilled, where $N(v)$ is the open neighborhood of $v$. The {em weight} of a 2RDF $f$ is the value $omega(f)=sum_{vin V}|f (v)|$. The {em $2$-rainbow domination number} of a graph $G$, denoted by $gamma_{r2}(G)$, is the minimum weight of a 2RDF of G. The {em annihilation number} $a(G)$ is the largest integer $k$ such that the sum of the first $k$ terms of the non-decreasing degree sequence of $G$ is at most the number of edges in $G$. In this paper, we prove that for any tree $T$ with at least two vertices, $gamma_{r2}(T)le a(T)+1$.
topic annihilation number
2-rainbow dominating function
2-rainbow domination number
url http://www.combinatorics.ir/?_action=showPDF&article=3051&_ob=dc39b3b99937a3eea4c41cc51272e53a&fileName=full_text.pdf.
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