Pebbling in Split Graphs

Autores
Alcón, Liliana Graciela; Gutierrez, Marisa; Hurlbert, Glenn
Año de publicación
2014
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
Graph pebbling is a network optimization model for transporting discrete resourcesthat are consumed in transit: the movement of 2 pebbles across an edge consumes one of thepebbles. The pebbling number of a graph is the fewest number of pebblestso that, from anyinitial configuration oftpebbles on its vertices, one can place a pebble on any given target vertex viasuch pebbling steps. It is known that deciding whether a given configuration on a particular graphcan reach a specified target isNP-complete, even for diameter 2 graphs, and that deciding whetherthe pebbling number has a prescribed upper bound is ΠP2-complete. On the other hand, for manyfamilies of graphs there are formulas or polynomial algorithms for computing pebbling numbers; forexample, complete graphs, products of paths (including cubes), trees, cycles, diameter 2 graphs, andmore. Moreover, graphs having minimum pebbling number are called Class 0, and many authors havestudied which graphs are Class 0 and what graph properties guarantee it, with no characterizationin sight. In this paper we investigate an important family of diameter 3 chordal graphs called splitgraphs; graphs whose vertex set can be partitioned into a clique and an independent set. We provide aformula for the pebbling number of a split graph, along with an algorithm for calculating it that runsinO(nβ) time, whereβ=2ω/(ω+1)∼=1.41 andω∼=2.376 is the exponent of matrix multiplication.Furthermore we determine that all split graphs with minimum degree at least 3 are Class 0.
Fil: Alcón, Liliana Graciela. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentina
Fil: Gutierrez, Marisa. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentina
Fil: Hurlbert, Glenn. Arizona State University; Estados Unidos
Materia
Pebbling Number
Split Graphs
Class 0
Graph Algorithms
Complexity
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:11336/32989
Acceder
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network_name_str CONICET Digital (CONICET)
spelling Pebbling in Split GraphsAlcón, Liliana GracielaGutierrez, MarisaHurlbert, GlennPebbling NumberSplit GraphsClass 0Graph AlgorithmsComplexityhttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1Graph pebbling is a network optimization model for transporting discrete resourcesthat are consumed in transit: the movement of 2 pebbles across an edge consumes one of thepebbles. The pebbling number of a graph is the fewest number of pebblestso that, from anyinitial configuration oftpebbles on its vertices, one can place a pebble on any given target vertex viasuch pebbling steps. It is known that deciding whether a given configuration on a particular graphcan reach a specified target isNP-complete, even for diameter 2 graphs, and that deciding whetherthe pebbling number has a prescribed upper bound is ΠP2-complete. On the other hand, for manyfamilies of graphs there are formulas or polynomial algorithms for computing pebbling numbers; forexample, complete graphs, products of paths (including cubes), trees, cycles, diameter 2 graphs, andmore. Moreover, graphs having minimum pebbling number are called Class 0, and many authors havestudied which graphs are Class 0 and what graph properties guarantee it, with no characterizationin sight. In this paper we investigate an important family of diameter 3 chordal graphs called splitgraphs; graphs whose vertex set can be partitioned into a clique and an independent set. We provide aformula for the pebbling number of a split graph, along with an algorithm for calculating it that runsinO(nβ) time, whereβ=2ω/(ω+1)∼=1.41 andω∼=2.376 is the exponent of matrix multiplication.Furthermore we determine that all split graphs with minimum degree at least 3 are Class 0.Fil: Alcón, Liliana Graciela. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; ArgentinaFil: Gutierrez, Marisa. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; ArgentinaFil: Hurlbert, Glenn. Arizona State University; Estados UnidosSociety for Industrial and Applied Mathematics2014-08info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/32989Hurlbert, Glenn; Gutierrez, Marisa; Alcón, Liliana Graciela; Pebbling in Split Graphs; Society for Industrial and Applied Mathematics; Siam Journal On Discrete Mathematics; 28; 3; 8-2014; 1449-14660895-4801CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/doi/10.1137/130914607info:eu-repo/semantics/altIdentifier/url/http://epubs.siam.org/doi/10.1137/130914607info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T10:02:52Zoai:ri.conicet.gov.ar:11336/32989instacron: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:34982025-09-29 10:02:52.445CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Pebbling in Split Graphs
title Pebbling in Split Graphs
spellingShingle Pebbling in Split Graphs
Alcón, Liliana Graciela
Pebbling Number
Split Graphs
Class 0
Graph Algorithms
Complexity
title_short Pebbling in Split Graphs
title_full Pebbling in Split Graphs
title_fullStr Pebbling in Split Graphs
title_full_unstemmed Pebbling in Split Graphs
title_sort Pebbling in Split Graphs
dc.creator.none.fl_str_mv Alcón, Liliana Graciela
Gutierrez, Marisa
Hurlbert, Glenn
author Alcón, Liliana Graciela
author_facet Alcón, Liliana Graciela
Gutierrez, Marisa
Hurlbert, Glenn
author_role author
author2 Gutierrez, Marisa
Hurlbert, Glenn
author2_role author
author
dc.subject.none.fl_str_mv Pebbling Number
Split Graphs
Class 0
Graph Algorithms
Complexity
topic Pebbling Number
Split Graphs
Class 0
Graph Algorithms
Complexity
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv Graph pebbling is a network optimization model for transporting discrete resourcesthat are consumed in transit: the movement of 2 pebbles across an edge consumes one of thepebbles. The pebbling number of a graph is the fewest number of pebblestso that, from anyinitial configuration oftpebbles on its vertices, one can place a pebble on any given target vertex viasuch pebbling steps. It is known that deciding whether a given configuration on a particular graphcan reach a specified target isNP-complete, even for diameter 2 graphs, and that deciding whetherthe pebbling number has a prescribed upper bound is ΠP2-complete. On the other hand, for manyfamilies of graphs there are formulas or polynomial algorithms for computing pebbling numbers; forexample, complete graphs, products of paths (including cubes), trees, cycles, diameter 2 graphs, andmore. Moreover, graphs having minimum pebbling number are called Class 0, and many authors havestudied which graphs are Class 0 and what graph properties guarantee it, with no characterizationin sight. In this paper we investigate an important family of diameter 3 chordal graphs called splitgraphs; graphs whose vertex set can be partitioned into a clique and an independent set. We provide aformula for the pebbling number of a split graph, along with an algorithm for calculating it that runsinO(nβ) time, whereβ=2ω/(ω+1)∼=1.41 andω∼=2.376 is the exponent of matrix multiplication.Furthermore we determine that all split graphs with minimum degree at least 3 are Class 0.
Fil: Alcón, Liliana Graciela. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentina
Fil: Gutierrez, Marisa. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentina
Fil: Hurlbert, Glenn. Arizona State University; Estados Unidos
description Graph pebbling is a network optimization model for transporting discrete resourcesthat are consumed in transit: the movement of 2 pebbles across an edge consumes one of thepebbles. The pebbling number of a graph is the fewest number of pebblestso that, from anyinitial configuration oftpebbles on its vertices, one can place a pebble on any given target vertex viasuch pebbling steps. It is known that deciding whether a given configuration on a particular graphcan reach a specified target isNP-complete, even for diameter 2 graphs, and that deciding whetherthe pebbling number has a prescribed upper bound is ΠP2-complete. On the other hand, for manyfamilies of graphs there are formulas or polynomial algorithms for computing pebbling numbers; forexample, complete graphs, products of paths (including cubes), trees, cycles, diameter 2 graphs, andmore. Moreover, graphs having minimum pebbling number are called Class 0, and many authors havestudied which graphs are Class 0 and what graph properties guarantee it, with no characterizationin sight. In this paper we investigate an important family of diameter 3 chordal graphs called splitgraphs; graphs whose vertex set can be partitioned into a clique and an independent set. We provide aformula for the pebbling number of a split graph, along with an algorithm for calculating it that runsinO(nβ) time, whereβ=2ω/(ω+1)∼=1.41 andω∼=2.376 is the exponent of matrix multiplication.Furthermore we determine that all split graphs with minimum degree at least 3 are Class 0.
publishDate 2014
dc.date.none.fl_str_mv 2014-08
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/11336/32989
Hurlbert, Glenn; Gutierrez, Marisa; Alcón, Liliana Graciela; Pebbling in Split Graphs; Society for Industrial and Applied Mathematics; Siam Journal On Discrete Mathematics; 28; 3; 8-2014; 1449-1466
0895-4801
CONICET Digital
CONICET
url http://hdl.handle.net/11336/32989
identifier_str_mv Hurlbert, Glenn; Gutierrez, Marisa; Alcón, Liliana Graciela; Pebbling in Split Graphs; Society for Industrial and Applied Mathematics; Siam Journal On Discrete Mathematics; 28; 3; 8-2014; 1449-1466
0895-4801
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/doi/10.1137/130914607
info:eu-repo/semantics/altIdentifier/url/http://epubs.siam.org/doi/10.1137/130914607
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
application/pdf
application/pdf
dc.publisher.none.fl_str_mv Society for Industrial and Applied Mathematics
publisher.none.fl_str_mv Society for Industrial and Applied Mathematics
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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