b-Coloring is NP-hard on Co-bipartite Graphs and Polytime Solvable on Tree Cographs

Autores
Bonomo, Flavia; Schaudt, Oliver; Stein, Maya; Valencia Pabon, Mario
Año de publicación
2015
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
A b-coloring of a graph is a proper coloring such that every color class contains a vertex that is adjacent to all other color classes. The b-chromatic number of a graph G, denoted by χb(G), is the maximum number t such that G admits a b-coloring with t colors. A graph G is called b-continuous if it admits a b-coloring with t colors, for every t = χ (G), . . . , χb(G), and b-monotonic if χb(H1) ≥ χb(H2) for every induced subgraph H1 of G, and every induced subgraph H2 of H1. We investigate the b-chromatic number of graphs with stability number two. These are exactly the complements of triangle-free graphs, thus including all complements of bipartite graphs. The main results of this work are the following: (1) We characterize the b-colorings of a graph with stability number two in terms of matchings with no augmenting paths of length one or three. We derive that graphs with stability number two are b-continuous and b-monotonic. (2) We prove that it is NP-complete to decide whether the b-chromatic number of co-bipartite graph is at least a given threshold. (3) We describe a polynomial time dynamic programming algorithm to compute the b-chromatic number of co-trees. (4) Extending several previous results, we show that there is a polynomial time dynamic programming algorithm for computing the b-chromatic number of tree-cographs. Moreover, we show that tree-cographs are b-continuous and b-monotonic.
Fil: Bonomo, Flavia. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina
Fil: Schaudt, Oliver. Universite Pierre et Marie Curie; Francia
Fil: Stein, Maya. Universidad de Chile; Chile
Fil: Valencia Pabon, Mario. Universite de Paris 13-Nord; Francia
Materia
B-Coloring
Minimum Maxinal Matching
Co-Bipartite Graphs
Graphs with Stability Number Two
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/18897
Acceder
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repository_id_str 3498
network_name_str CONICET Digital (CONICET)
spelling b-Coloring is NP-hard on Co-bipartite Graphs and Polytime Solvable on Tree CographsBonomo, FlaviaSchaudt, OliverStein, MayaValencia Pabon, MarioB-ColoringMinimum Maxinal MatchingCo-Bipartite GraphsGraphs with Stability Number Twohttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1https://purl.org/becyt/ford/1.2https://purl.org/becyt/ford/1A b-coloring of a graph is a proper coloring such that every color class contains a vertex that is adjacent to all other color classes. The b-chromatic number of a graph G, denoted by χb(G), is the maximum number t such that G admits a b-coloring with t colors. A graph G is called b-continuous if it admits a b-coloring with t colors, for every t = χ (G), . . . , χb(G), and b-monotonic if χb(H1) ≥ χb(H2) for every induced subgraph H1 of G, and every induced subgraph H2 of H1. We investigate the b-chromatic number of graphs with stability number two. These are exactly the complements of triangle-free graphs, thus including all complements of bipartite graphs. The main results of this work are the following: (1) We characterize the b-colorings of a graph with stability number two in terms of matchings with no augmenting paths of length one or three. We derive that graphs with stability number two are b-continuous and b-monotonic. (2) We prove that it is NP-complete to decide whether the b-chromatic number of co-bipartite graph is at least a given threshold. (3) We describe a polynomial time dynamic programming algorithm to compute the b-chromatic number of co-trees. (4) Extending several previous results, we show that there is a polynomial time dynamic programming algorithm for computing the b-chromatic number of tree-cographs. Moreover, we show that tree-cographs are b-continuous and b-monotonic.Fil: Bonomo, Flavia. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Schaudt, Oliver. Universite Pierre et Marie Curie; FranciaFil: Stein, Maya. Universidad de Chile; ChileFil: Valencia Pabon, Mario. Universite de Paris 13-Nord; FranciaSpringer2015-10info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/18897Bonomo, Flavia; Schaudt, Oliver; Stein, Maya; Valencia Pabon, Mario; b-Coloring is NP-hard on Co-bipartite Graphs and Polytime Solvable on Tree Cographs; Springer; Algorithmica; 73; 2; 10-2015; 289-3050178-46171432-0541CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/doi/10.1007/s00453-014-9921-5info:eu-repo/semantics/altIdentifier/url/https://link.springer.com/article/10.1007%2Fs00453-014-9921-5info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1310.8313info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-10-15T15:13:16Zoai:ri.conicet.gov.ar:11336/18897instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-10-15 15:13:16.394CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv b-Coloring is NP-hard on Co-bipartite Graphs and Polytime Solvable on Tree Cographs
title b-Coloring is NP-hard on Co-bipartite Graphs and Polytime Solvable on Tree Cographs
spellingShingle b-Coloring is NP-hard on Co-bipartite Graphs and Polytime Solvable on Tree Cographs
Bonomo, Flavia
B-Coloring
Minimum Maxinal Matching
Co-Bipartite Graphs
Graphs with Stability Number Two
title_short b-Coloring is NP-hard on Co-bipartite Graphs and Polytime Solvable on Tree Cographs
title_full b-Coloring is NP-hard on Co-bipartite Graphs and Polytime Solvable on Tree Cographs
title_fullStr b-Coloring is NP-hard on Co-bipartite Graphs and Polytime Solvable on Tree Cographs
title_full_unstemmed b-Coloring is NP-hard on Co-bipartite Graphs and Polytime Solvable on Tree Cographs
title_sort b-Coloring is NP-hard on Co-bipartite Graphs and Polytime Solvable on Tree Cographs
dc.creator.none.fl_str_mv Bonomo, Flavia
Schaudt, Oliver
Stein, Maya
Valencia Pabon, Mario
author Bonomo, Flavia
author_facet Bonomo, Flavia
Schaudt, Oliver
Stein, Maya
Valencia Pabon, Mario
author_role author
author2 Schaudt, Oliver
Stein, Maya
Valencia Pabon, Mario
author2_role author
author
author
dc.subject.none.fl_str_mv B-Coloring
Minimum Maxinal Matching
Co-Bipartite Graphs
Graphs with Stability Number Two
topic B-Coloring
Minimum Maxinal Matching
Co-Bipartite Graphs
Graphs with Stability Number Two
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
https://purl.org/becyt/ford/1.2
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv A b-coloring of a graph is a proper coloring such that every color class contains a vertex that is adjacent to all other color classes. The b-chromatic number of a graph G, denoted by χb(G), is the maximum number t such that G admits a b-coloring with t colors. A graph G is called b-continuous if it admits a b-coloring with t colors, for every t = χ (G), . . . , χb(G), and b-monotonic if χb(H1) ≥ χb(H2) for every induced subgraph H1 of G, and every induced subgraph H2 of H1. We investigate the b-chromatic number of graphs with stability number two. These are exactly the complements of triangle-free graphs, thus including all complements of bipartite graphs. The main results of this work are the following: (1) We characterize the b-colorings of a graph with stability number two in terms of matchings with no augmenting paths of length one or three. We derive that graphs with stability number two are b-continuous and b-monotonic. (2) We prove that it is NP-complete to decide whether the b-chromatic number of co-bipartite graph is at least a given threshold. (3) We describe a polynomial time dynamic programming algorithm to compute the b-chromatic number of co-trees. (4) Extending several previous results, we show that there is a polynomial time dynamic programming algorithm for computing the b-chromatic number of tree-cographs. Moreover, we show that tree-cographs are b-continuous and b-monotonic.
Fil: Bonomo, Flavia. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina
Fil: Schaudt, Oliver. Universite Pierre et Marie Curie; Francia
Fil: Stein, Maya. Universidad de Chile; Chile
Fil: Valencia Pabon, Mario. Universite de Paris 13-Nord; Francia
description A b-coloring of a graph is a proper coloring such that every color class contains a vertex that is adjacent to all other color classes. The b-chromatic number of a graph G, denoted by χb(G), is the maximum number t such that G admits a b-coloring with t colors. A graph G is called b-continuous if it admits a b-coloring with t colors, for every t = χ (G), . . . , χb(G), and b-monotonic if χb(H1) ≥ χb(H2) for every induced subgraph H1 of G, and every induced subgraph H2 of H1. We investigate the b-chromatic number of graphs with stability number two. These are exactly the complements of triangle-free graphs, thus including all complements of bipartite graphs. The main results of this work are the following: (1) We characterize the b-colorings of a graph with stability number two in terms of matchings with no augmenting paths of length one or three. We derive that graphs with stability number two are b-continuous and b-monotonic. (2) We prove that it is NP-complete to decide whether the b-chromatic number of co-bipartite graph is at least a given threshold. (3) We describe a polynomial time dynamic programming algorithm to compute the b-chromatic number of co-trees. (4) Extending several previous results, we show that there is a polynomial time dynamic programming algorithm for computing the b-chromatic number of tree-cographs. Moreover, we show that tree-cographs are b-continuous and b-monotonic.
publishDate 2015
dc.date.none.fl_str_mv 2015-10
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11336/18897
Bonomo, Flavia; Schaudt, Oliver; Stein, Maya; Valencia Pabon, Mario; b-Coloring is NP-hard on Co-bipartite Graphs and Polytime Solvable on Tree Cographs; Springer; Algorithmica; 73; 2; 10-2015; 289-305
0178-4617
1432-0541
CONICET Digital
CONICET
url http://hdl.handle.net/11336/18897
identifier_str_mv Bonomo, Flavia; Schaudt, Oliver; Stein, Maya; Valencia Pabon, Mario; b-Coloring is NP-hard on Co-bipartite Graphs and Polytime Solvable on Tree Cographs; Springer; Algorithmica; 73; 2; 10-2015; 289-305
0178-4617
1432-0541
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/doi/10.1007/s00453-014-9921-5
info:eu-repo/semantics/altIdentifier/url/https://link.springer.com/article/10.1007%2Fs00453-014-9921-5
info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1310.8313
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv Springer
publisher.none.fl_str_mv Springer
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
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score 13.22299

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