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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University of Isfahan
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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. |
work_keys_str_mv |
AT nasrindehgardi boundingtherainbowdominationnumberofatreeintermsofitsannihilationnumber AT mahmoudsheikholeslami boundingtherainbowdominationnumberofatreeintermsofitsannihilationnumber AT abdollahkhodkar boundingtherainbowdominationnumberofatreeintermsofitsannihilationnumber |
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1725425350805028864 |