Partitioning chordal graphs into independent sets and cliques

Detalhes bibliográficos
Autor(a) principal: Hell, Pavol
Data de Publicação: 2001
Outros Autores: Klein, Sulamita, Nogueira, Loana Tito, Protti, Fábio
Tipo de documento: Relatório
Idioma: eng
Título da fonte: Repositório Institucional da UFRJ
Texto Completo: http://hdl.handle.net/11422/1889
Resumo: We consider the following generalization of split graphs: A graph is said to be a (k,ℓ)-graph if its vertex set can be partitioned into k independent sets and ℓ cliques. (Split graphs are obtained by setting k=ℓ=1.) Much of the appeal of split graphs is due to the fact that they are chordal, a property not shared by (k,ℓ)-graphs in general. (For instance, being a (k,0)-graph is equivalent to being k-colourable.) However, if we keep the assumption of chordality, nice algorithms and characterization theorems are possible. Indeed, our main result is a forbidden subgraph characterization of chordal (k,ℓ)-graphs. We also give an O(n(m+n)) recognition algorithm for chordal (k,ℓ)-graphs. When k=1, our algorithm runs in time O(m+n). In particular, we obtain a new simple and efficient greedy algorithm for the recognition of split graphs, from which it is easy to derive the well known forbidden subgraph characterization of split graphs. The algorithm and the characterization extend, in a natural way, to the ‘list’ (or ‘pre-colouring extension’) version of the split partition problem — given a graph with some vertices pre-assigned to the independent set, or to the clique, is there a split partition extending this pre-assignment? Another way to think of our main result is the following min-max property of chordal graphs: the maximum number of independent (i.e., disjoint and nonadjacent) Kr's equals the minimum number of cliques that meet all Kr's.
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spelling Partitioning chordal graphs into independent sets and cliquesChordal graphsSplit graphsGreedy algorithmsList partitionsMin-max theoremsGrafos cordalGrafos cliqueCNPQ::CIENCIAS EXATAS E DA TERRA::CIENCIA DA COMPUTACAOWe consider the following generalization of split graphs: A graph is said to be a (k,ℓ)-graph if its vertex set can be partitioned into k independent sets and ℓ cliques. (Split graphs are obtained by setting k=ℓ=1.) Much of the appeal of split graphs is due to the fact that they are chordal, a property not shared by (k,ℓ)-graphs in general. (For instance, being a (k,0)-graph is equivalent to being k-colourable.) However, if we keep the assumption of chordality, nice algorithms and characterization theorems are possible. Indeed, our main result is a forbidden subgraph characterization of chordal (k,ℓ)-graphs. We also give an O(n(m+n)) recognition algorithm for chordal (k,ℓ)-graphs. When k=1, our algorithm runs in time O(m+n). In particular, we obtain a new simple and efficient greedy algorithm for the recognition of split graphs, from which it is easy to derive the well known forbidden subgraph characterization of split graphs. The algorithm and the characterization extend, in a natural way, to the ‘list’ (or ‘pre-colouring extension’) version of the split partition problem — given a graph with some vertices pre-assigned to the independent set, or to the clique, is there a split partition extending this pre-assignment? Another way to think of our main result is the following min-max property of chordal graphs: the maximum number of independent (i.e., disjoint and nonadjacent) Kr's equals the minimum number of cliques that meet all Kr's.BrasilInstituto Tércio Pacitti de Aplicações e Pesquisas Computacionais2017-05-08T15:01:43Z2023-12-21T03:00:54Z2001-12-31info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/reportHELL, P. et al. Partitioning chordal graphs into independent sets and cliques. Rio de Janeiro: NCE/UFRJ, 2001. (Relatório Técnico, 05/01)http://hdl.handle.net/11422/1889engRelatório Técnico NCEHell, PavolKlein, SulamitaNogueira, Loana TitoProtti, Fábioinfo:eu-repo/semantics/openAccessreponame:Repositório Institucional da UFRJinstname:Universidade Federal do Rio de Janeiro (UFRJ)instacron:UFRJ2023-12-21T03:00:54Zoai:pantheon.ufrj.br:11422/1889Repositório InstitucionalPUBhttp://www.pantheon.ufrj.br/oai/requestpantheon@sibi.ufrj.bropendoar:2023-12-21T03:00:54Repositório Institucional da UFRJ - Universidade Federal do Rio de Janeiro (UFRJ)false
dc.title.none.fl_str_mv Partitioning chordal graphs into independent sets and cliques
title Partitioning chordal graphs into independent sets and cliques
spellingShingle Partitioning chordal graphs into independent sets and cliques
Hell, Pavol
Chordal graphs
Split graphs
Greedy algorithms
List partitions
Min-max theorems
Grafos cordal
Grafos clique
CNPQ::CIENCIAS EXATAS E DA TERRA::CIENCIA DA COMPUTACAO
title_short Partitioning chordal graphs into independent sets and cliques
title_full Partitioning chordal graphs into independent sets and cliques
title_fullStr Partitioning chordal graphs into independent sets and cliques
title_full_unstemmed Partitioning chordal graphs into independent sets and cliques
title_sort Partitioning chordal graphs into independent sets and cliques
author Hell, Pavol
author_facet Hell, Pavol
Klein, Sulamita
Nogueira, Loana Tito
Protti, Fábio
author_role author
author2 Klein, Sulamita
Nogueira, Loana Tito
Protti, Fábio
author2_role author
author
author
dc.contributor.author.fl_str_mv Hell, Pavol
Klein, Sulamita
Nogueira, Loana Tito
Protti, Fábio
dc.subject.por.fl_str_mv Chordal graphs
Split graphs
Greedy algorithms
List partitions
Min-max theorems
Grafos cordal
Grafos clique
CNPQ::CIENCIAS EXATAS E DA TERRA::CIENCIA DA COMPUTACAO
topic Chordal graphs
Split graphs
Greedy algorithms
List partitions
Min-max theorems
Grafos cordal
Grafos clique
CNPQ::CIENCIAS EXATAS E DA TERRA::CIENCIA DA COMPUTACAO
description We consider the following generalization of split graphs: A graph is said to be a (k,ℓ)-graph if its vertex set can be partitioned into k independent sets and ℓ cliques. (Split graphs are obtained by setting k=ℓ=1.) Much of the appeal of split graphs is due to the fact that they are chordal, a property not shared by (k,ℓ)-graphs in general. (For instance, being a (k,0)-graph is equivalent to being k-colourable.) However, if we keep the assumption of chordality, nice algorithms and characterization theorems are possible. Indeed, our main result is a forbidden subgraph characterization of chordal (k,ℓ)-graphs. We also give an O(n(m+n)) recognition algorithm for chordal (k,ℓ)-graphs. When k=1, our algorithm runs in time O(m+n). In particular, we obtain a new simple and efficient greedy algorithm for the recognition of split graphs, from which it is easy to derive the well known forbidden subgraph characterization of split graphs. The algorithm and the characterization extend, in a natural way, to the ‘list’ (or ‘pre-colouring extension’) version of the split partition problem — given a graph with some vertices pre-assigned to the independent set, or to the clique, is there a split partition extending this pre-assignment? Another way to think of our main result is the following min-max property of chordal graphs: the maximum number of independent (i.e., disjoint and nonadjacent) Kr's equals the minimum number of cliques that meet all Kr's.
publishDate 2001
dc.date.none.fl_str_mv 2001-12-31
2017-05-08T15:01:43Z
2023-12-21T03:00:54Z
dc.type.status.fl_str_mv info:eu-repo/semantics/publishedVersion
dc.type.driver.fl_str_mv info:eu-repo/semantics/report
format report
status_str publishedVersion
dc.identifier.uri.fl_str_mv HELL, P. et al. Partitioning chordal graphs into independent sets and cliques. Rio de Janeiro: NCE/UFRJ, 2001. (Relatório Técnico, 05/01)
http://hdl.handle.net/11422/1889
identifier_str_mv HELL, P. et al. Partitioning chordal graphs into independent sets and cliques. Rio de Janeiro: NCE/UFRJ, 2001. (Relatório Técnico, 05/01)
url http://hdl.handle.net/11422/1889
dc.language.iso.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv Relatório Técnico NCE
dc.rights.driver.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.publisher.none.fl_str_mv Brasil
Instituto Tércio Pacitti de Aplicações e Pesquisas Computacionais
publisher.none.fl_str_mv Brasil
Instituto Tércio Pacitti de Aplicações e Pesquisas Computacionais
dc.source.none.fl_str_mv reponame:Repositório Institucional da UFRJ
instname:Universidade Federal do Rio de Janeiro (UFRJ)
instacron:UFRJ
instname_str Universidade Federal do Rio de Janeiro (UFRJ)
instacron_str UFRJ
institution UFRJ
reponame_str Repositório Institucional da UFRJ
collection Repositório Institucional da UFRJ
repository.name.fl_str_mv Repositório Institucional da UFRJ - Universidade Federal do Rio de Janeiro (UFRJ)
repository.mail.fl_str_mv pantheon@sibi.ufrj.br
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