Rebuilding convex sets in graphs

The usual distance between pairs of vertices in a graph naturally gives rise to the notion of an interval between a pair of vertices in a graph. This in turn allows us to extend the notions of convex sets, convex hull, and extreme points in Euclidean space to the vertex set of a graph. The extreme v...

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Autores: Cáceres González, José, Márquez Pérez, Alberto, Oellermann, Ortrud R., Puertas González, María Luz
Tipo de recurso: artículo
Estado:Versión publicada
Fecha de publicación:2005
País:España
Institución:Universidad de Sevilla (US)
Repositorio:idUS. Depósito de Investigación de la Universidad de Sevilla
OAI Identifier:oai:idus.us.es:11441/34391
Acceso en línea:http://hdl.handle.net/11441/34391
https://doi.org/10.1016/j.disc.2005.03.020
Access Level:acceso abierto
Palabra clave:Eccentricity
Contour vertex
Distance hereditary graph
Convex hull
Geodetic set
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spelling Rebuilding convex sets in graphsCáceres González, JoséMárquez Pérez, AlbertoOellermann, Ortrud R.Puertas González, María LuzEccentricityContour vertexDistance hereditary graphConvex hullGeodetic setThe usual distance between pairs of vertices in a graph naturally gives rise to the notion of an interval between a pair of vertices in a graph. This in turn allows us to extend the notions of convex sets, convex hull, and extreme points in Euclidean space to the vertex set of a graph. The extreme vertices of a graph are known to be precisely the simplicial vertices, i.e., the vertices whose neighborhoods are complete graphs. It is known that the class of graphs with the Minkowski–Krein–Milman property, i.e., the property that every convex set is the convex hull of its extreme points, is precisely the class of chordal graphs without induced 3-fans. We define a vertex to be a contour vertex if the eccentricity of every neighbor is at most as large as that of the vertex. In this paper we show that every convex set of vertices in a graph is the convex hull of the collection of its contour vertices. We characterize those graphs for which every convex set has the property that its contour vertices coincide with its extreme points. A set of vertices in a graph is a geodetic set if the union of the intervals between pairs of vertices in the set, taken over all pairs in the set, is the entire vertex set. We show that the contour vertices in distance hereditary graphs form a geodetic set.Matemática Aplicada I2005info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionapplication/pdfapplication/pdfhttp://hdl.handle.net/11441/34391https://doi.org/10.1016/j.disc.2005.03.020reponame:idUS. Depósito de Investigación de la Universidad de Sevillainstname:Universidad de Sevilla (US)InglésDiscrete Mathematics, 297 (1-3), 26-37.info:eu-repo/semantics/openAccessoai:idus.us.es:11441/343912026-06-17T12:51:07Z
dc.title.none.fl_str_mv Rebuilding convex sets in graphs
title Rebuilding convex sets in graphs
spellingShingle Rebuilding convex sets in graphs
Cáceres González, José
Eccentricity
Contour vertex
Distance hereditary graph
Convex hull
Geodetic set
title_short Rebuilding convex sets in graphs
title_full Rebuilding convex sets in graphs
title_fullStr Rebuilding convex sets in graphs
title_full_unstemmed Rebuilding convex sets in graphs
title_sort Rebuilding convex sets in graphs
dc.creator.none.fl_str_mv Cáceres González, José
Márquez Pérez, Alberto
Oellermann, Ortrud R.
Puertas González, María Luz
author Cáceres González, José
author_facet Cáceres González, José
Márquez Pérez, Alberto
Oellermann, Ortrud R.
Puertas González, María Luz
author_role author
author2 Márquez Pérez, Alberto
Oellermann, Ortrud R.
Puertas González, María Luz
author2_role author
author
author
dc.contributor.none.fl_str_mv Matemática Aplicada I
dc.subject.none.fl_str_mv Eccentricity
Contour vertex
Distance hereditary graph
Convex hull
Geodetic set
topic Eccentricity
Contour vertex
Distance hereditary graph
Convex hull
Geodetic set
description The usual distance between pairs of vertices in a graph naturally gives rise to the notion of an interval between a pair of vertices in a graph. This in turn allows us to extend the notions of convex sets, convex hull, and extreme points in Euclidean space to the vertex set of a graph. The extreme vertices of a graph are known to be precisely the simplicial vertices, i.e., the vertices whose neighborhoods are complete graphs. It is known that the class of graphs with the Minkowski–Krein–Milman property, i.e., the property that every convex set is the convex hull of its extreme points, is precisely the class of chordal graphs without induced 3-fans. We define a vertex to be a contour vertex if the eccentricity of every neighbor is at most as large as that of the vertex. In this paper we show that every convex set of vertices in a graph is the convex hull of the collection of its contour vertices. We characterize those graphs for which every convex set has the property that its contour vertices coincide with its extreme points. A set of vertices in a graph is a geodetic set if the union of the intervals between pairs of vertices in the set, taken over all pairs in the set, is the entire vertex set. We show that the contour vertices in distance hereditary graphs form a geodetic set.
publishDate 2005
dc.date.none.fl_str_mv 2005
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11441/34391
https://doi.org/10.1016/j.disc.2005.03.020
url http://hdl.handle.net/11441/34391
https://doi.org/10.1016/j.disc.2005.03.020
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Discrete Mathematics, 297 (1-3), 26-37.
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.source.none.fl_str_mv reponame:idUS. Depósito de Investigación de la Universidad de Sevilla
instname:Universidad de Sevilla (US)
instname_str Universidad de Sevilla (US)
reponame_str idUS. Depósito de Investigación de la Universidad de Sevilla
collection idUS. Depósito de Investigación de la Universidad de Sevilla
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