Convex sets in graphs. II: Minimal path convexity
type: Item
instance of: scholarly article
title: Convex sets in graphs. II: Minimal path convexity
zbMATH Open document ID: 0672.52001
DOI: 10.1016/0095-8956(88)90039-1
author: Pierre Duchet
published in: Journal of Combinatorial Theory. Series B
publication date: 1988-01-01T00:00:00Z
review text: [For part I see the author and \textit{H. Meyniel}, Eur. J. Comb. 4, 127-132 (1983; Zbl 0523.05031).] The basic structure is a graph-convexity space (G,\({\mathcal C})\) with G a connected graph with vertex set V, and \({\mathcal C}^ a \)convexity structure on V such that (a) \(\emptyset,V\in {\mathcal C}\), (b) \({\mathcal C}\) is closed under arbitrary intersections, (c) \({\mathcal C}\) is closed under nested unions, and (d) every member of \({\mathcal C}\) induces a connected subgraph of G. The minimal path convexity can be defined by an interval function I: \(V\times V\to 2^ V\) with I(x,y) the set of all vertices of all chordless (x,y)-paths; the elements of \({\mathcal C}\) then are those sets which are closed under the operator I. The goal of this paper is to show how particular minimal path convexity spaces are by determining the exact values of the Carathéodory, Helly and Radon numbers.
Mathematics Subject Classification ID: 52A01
Mathematics Subject Classification ID: 52A35
Mathematics Subject Classification ID: 05C40
zbMATH DE Number: 4100058
zbMATH Keywords: Carathéodory number
zbMATH Keywords: Radon number
zbMATH Keywords: Helly number
zbMATH Keywords: graph-convexity
zbMATH Keywords: minimal path convexity spaces
MaRDI profile type: Publication
full work available at URL: 0095-8956(88)90039-1
OpenAlex ID: W2117475001
Wikidata QID: Q56503424
cites work: Helly and Radon-type theorems in interval convexity spaces
cites work: Q4088860
cites work: Convexity spaces. I: The basis properties
cites work: Some Elementary Properties of Interval Convexities
cites work: Sous Les Pavés…
cites work: Ensemble convexes dans les graphes. I: Théoremes de Helly et de Radon pour graphes et surfaces
cites work: Q4200096
cites work: Q3889702
cites work: The theory of convex geometries
cites work: Partition numbers for trees and ordered sets
cites work: Q3940405
cites work: A Helly theorem for convexity in graphs
cites work: Axiomatic convexity theory and relationships between the Carathéodory, Helly, and Radon numbers
cites work: Q5807392
cites work: Q3890733
cites work: Über die Rolle der transfiniten Schlußweisen in einer allgemeinen Idealtheorie
cites work: Q4125422
cites work: Q5185609
cites work: Q5338702
DBLP publication ID: journals/jct/Duchet88
Recommended article: Q3595105
Recommended article: Graphs with a minimal number of convex sets
Recommended article: On the convexity of paths of length two in undirected graphs
Recommended article: Q4949868
Recommended article: On finite convexity spaces induced by sets of paths in graphs
Recommended article: Q4034777
Recommended article: Convexities related to path properties on graphs
Recommended article: Q3815320
Recommended article: Convex combinations of minimal graphs
Recommended article: Q4519732
description: scientific article; zbMATH DE number 4100058