Recognizing vertex intersection graphs of paths on bounded degree trees

Autores
Alcón, Liliana Graciela; Gutiérrez, Marisa; Mazzoleni, María Pía
Año de publicación
2014
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
An (h,s,t)-representation of a graph G consists of a collection of subtrees of a tree T, where each subtree corresponds to a vertex of G such that (i) the maximum degree of T is at most h, (ii) every subtree has maximum degree at most s, (iii) there is an edge between two vertices in the graph G if and only if the corresponding subtrees have at least t vertices in common in T. The class of graphs that has an (h,s,t)-representation is denoted by [h,s,t]. An undirected graph G is called a VPT graph if it is the vertex intersection graph of a family of paths in a tree. Thus, [h,2,1] graphs are the VPT graphs that can be represented in a tree with maximum degree at most h. In this paper we characterize [h,2,1] graphs using chromatic number. We show that the problem of deciding whether a given VPT graph belongs to [h,2,1] is NP-complete, while the problem of deciding whether the graph belongs to [h,2,1]-[h-1,2,1] is NP-hard. Both problems remain hard even when restricted to VPT∩Split. Additionally, we present a non-trivial subclass of VPT∩Split in which these problems are polynomial time solvable.
Facultad de Ciencias Exactas
Materia
Matemática
Intersection graphs
Recognition problems
Representations on trees
VPT graphs
Nivel de accesibilidad
acceso abierto
Condiciones de uso
http://creativecommons.org/licenses/by-nc-sa/4.0/
Repositorio
SEDICI (UNLP)
Institución
Universidad Nacional de La Plata
OAI Identificador
oai:sedici.unlp.edu.ar:10915/85561

id SEDICI_71f6fef39e0d5562b62dc95324064f1c
oai_identifier_str oai:sedici.unlp.edu.ar:10915/85561
network_acronym_str SEDICI
repository_id_str 1329
network_name_str SEDICI (UNLP)
spelling Recognizing vertex intersection graphs of paths on bounded degree treesAlcón, Liliana GracielaGutiérrez, MarisaMazzoleni, María PíaMatemáticaIntersection graphsRecognition problemsRepresentations on treesVPT graphsAn (h,s,t)-representation of a graph G consists of a collection of subtrees of a tree T, where each subtree corresponds to a vertex of G such that (i) the maximum degree of T is at most h, (ii) every subtree has maximum degree at most s, (iii) there is an edge between two vertices in the graph G if and only if the corresponding subtrees have at least t vertices in common in T. The class of graphs that has an (h,s,t)-representation is denoted by [h,s,t]. An undirected graph G is called a VPT graph if it is the vertex intersection graph of a family of paths in a tree. Thus, [h,2,1] graphs are the VPT graphs that can be represented in a tree with maximum degree at most h. In this paper we characterize [h,2,1] graphs using chromatic number. We show that the problem of deciding whether a given VPT graph belongs to [h,2,1] is NP-complete, while the problem of deciding whether the graph belongs to [h,2,1]-[h-1,2,1] is NP-hard. Both problems remain hard even when restricted to VPT∩Split. Additionally, we present a non-trivial subclass of VPT∩Split in which these problems are polynomial time solvable.Facultad de Ciencias Exactas2014info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionArticulohttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdf70-77http://sedici.unlp.edu.ar/handle/10915/85561enginfo:eu-repo/semantics/altIdentifier/issn/0166-218Xinfo:eu-repo/semantics/altIdentifier/doi/10.1016/j.dam.2013.08.004info:eu-repo/semantics/openAccesshttp://creativecommons.org/licenses/by-nc-sa/4.0/Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0)reponame:SEDICI (UNLP)instname:Universidad Nacional de La Platainstacron:UNLP2025-09-03T10:48:41Zoai:sedici.unlp.edu.ar:10915/85561Institucionalhttp://sedici.unlp.edu.ar/Universidad públicaNo correspondehttp://sedici.unlp.edu.ar/oai/snrdalira@sedici.unlp.edu.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:13292025-09-03 10:48:41.37SEDICI (UNLP) - Universidad Nacional de La Platafalse
dc.title.none.fl_str_mv Recognizing vertex intersection graphs of paths on bounded degree trees
title Recognizing vertex intersection graphs of paths on bounded degree trees
spellingShingle Recognizing vertex intersection graphs of paths on bounded degree trees
Alcón, Liliana Graciela
Matemática
Intersection graphs
Recognition problems
Representations on trees
VPT graphs
title_short Recognizing vertex intersection graphs of paths on bounded degree trees
title_full Recognizing vertex intersection graphs of paths on bounded degree trees
title_fullStr Recognizing vertex intersection graphs of paths on bounded degree trees
title_full_unstemmed Recognizing vertex intersection graphs of paths on bounded degree trees
title_sort Recognizing vertex intersection graphs of paths on bounded degree trees
dc.creator.none.fl_str_mv Alcón, Liliana Graciela
Gutiérrez, Marisa
Mazzoleni, María Pía
author Alcón, Liliana Graciela
author_facet Alcón, Liliana Graciela
Gutiérrez, Marisa
Mazzoleni, María Pía
author_role author
author2 Gutiérrez, Marisa
Mazzoleni, María Pía
author2_role author
author
dc.subject.none.fl_str_mv Matemática
Intersection graphs
Recognition problems
Representations on trees
VPT graphs
topic Matemática
Intersection graphs
Recognition problems
Representations on trees
VPT graphs
dc.description.none.fl_txt_mv An (h,s,t)-representation of a graph G consists of a collection of subtrees of a tree T, where each subtree corresponds to a vertex of G such that (i) the maximum degree of T is at most h, (ii) every subtree has maximum degree at most s, (iii) there is an edge between two vertices in the graph G if and only if the corresponding subtrees have at least t vertices in common in T. The class of graphs that has an (h,s,t)-representation is denoted by [h,s,t]. An undirected graph G is called a VPT graph if it is the vertex intersection graph of a family of paths in a tree. Thus, [h,2,1] graphs are the VPT graphs that can be represented in a tree with maximum degree at most h. In this paper we characterize [h,2,1] graphs using chromatic number. We show that the problem of deciding whether a given VPT graph belongs to [h,2,1] is NP-complete, while the problem of deciding whether the graph belongs to [h,2,1]-[h-1,2,1] is NP-hard. Both problems remain hard even when restricted to VPT∩Split. Additionally, we present a non-trivial subclass of VPT∩Split in which these problems are polynomial time solvable.
Facultad de Ciencias Exactas
description An (h,s,t)-representation of a graph G consists of a collection of subtrees of a tree T, where each subtree corresponds to a vertex of G such that (i) the maximum degree of T is at most h, (ii) every subtree has maximum degree at most s, (iii) there is an edge between two vertices in the graph G if and only if the corresponding subtrees have at least t vertices in common in T. The class of graphs that has an (h,s,t)-representation is denoted by [h,s,t]. An undirected graph G is called a VPT graph if it is the vertex intersection graph of a family of paths in a tree. Thus, [h,2,1] graphs are the VPT graphs that can be represented in a tree with maximum degree at most h. In this paper we characterize [h,2,1] graphs using chromatic number. We show that the problem of deciding whether a given VPT graph belongs to [h,2,1] is NP-complete, while the problem of deciding whether the graph belongs to [h,2,1]-[h-1,2,1] is NP-hard. Both problems remain hard even when restricted to VPT∩Split. Additionally, we present a non-trivial subclass of VPT∩Split in which these problems are polynomial time solvable.
publishDate 2014
dc.date.none.fl_str_mv 2014
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
Articulo
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://sedici.unlp.edu.ar/handle/10915/85561
url http://sedici.unlp.edu.ar/handle/10915/85561
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/issn/0166-218X
info:eu-repo/semantics/altIdentifier/doi/10.1016/j.dam.2013.08.004
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc-sa/4.0/
Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0)
eu_rights_str_mv openAccess
rights_invalid_str_mv http://creativecommons.org/licenses/by-nc-sa/4.0/
Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0)
dc.format.none.fl_str_mv application/pdf
70-77
dc.source.none.fl_str_mv reponame:SEDICI (UNLP)
instname:Universidad Nacional de La Plata
instacron:UNLP
reponame_str SEDICI (UNLP)
collection SEDICI (UNLP)
instname_str Universidad Nacional de La Plata
instacron_str UNLP
institution UNLP
repository.name.fl_str_mv SEDICI (UNLP) - Universidad Nacional de La Plata
repository.mail.fl_str_mv alira@sedici.unlp.edu.ar
_version_ 1842260362594877440
score 13.13397