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
- Institución
- Universidad Nacional de La Plata
- OAI Identificador
- oai:sedici.unlp.edu.ar:10915/85561
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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 |
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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 |
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info:eu-repo/semantics/altIdentifier/issn/0166-218X info:eu-repo/semantics/altIdentifier/doi/10.1016/j.dam.2013.08.004 |
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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) |
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openAccess |
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http://creativecommons.org/licenses/by-nc-sa/4.0/ Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0) |
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