Minimax theorems for infinite graphs with the ends as ideal points
type: Item
instance of: scholarly article
title: Minimax theorems for infinite graphs with the ends as ideal points
zbMATH Open document ID: 0817.05064
DOI: 10.1016/0012-365X(92)00525-V
author: Norbert Polat
published in: Discrete Mathematics
publication date: 1995-07-23T00:00:00Z
review text: For a family \({\mathcal A}= (A_ i)_{i\in I}\) of disjoint sets of ends in a graph \(G= (V, E)\) the author proves Menger-type results involving the concepts of an \({\mathcal A}\)-path and of an \({\mathcal A}\)-separator. By definition, an \({\mathcal A}\)-path joins elements of distinct \(A_ i\) and \(A_ j\), while a path \(P\), finite or infinite, joins the distinct ends \(\tau_ 1\) and \(\tau_ 2\), if \(\tau_ h\) corresponds to a ray \(R_ h\subseteq P\) with \(R_ h\in \tau_ h\) or else to a terminal vertex \(v_ h\) of \(P\) which then cannot be separated from \(v_ h\) by a finite set \(F\subseteq V- \{v_ h\}\) (\(h= 1,2\); \(v_ 1\neq v_ 2\), if \(P\) is finite). A set \(M\) of vertices and edges separates \(\mathcal A\) if each \({\mathcal A}\)-path contains an element of \(M\). By definition, such a set is dispersed by \(\mathcal A\) if for each \(\tau\in \bigcap {\mathcal A}\) some finite set \(F\subseteq V\) separates \(M\) from \(\tau\) (in the usual sense). By the two main results the maximum cardinality of sets of disjoint \({\mathcal A}\)- arcs equals the minimum cardinality of certain \({\mathcal A}\)-separating sets which are dispersed by \(\mathcal A\). Special attention is given to the case when \(| I|= 2\).
Mathematics Subject Classification ID: 05C99
zbMATH DE Number: 626314
zbMATH Keywords: minimax theorems
zbMATH Keywords: infinite graphs
zbMATH Keywords: ends
zbMATH Keywords: path
reviewed by: Heinz A. Jung
MaRDI profile type: Publication
cites work: Über die Enden diskreter Räume und Gruppen
cites work: Über unendliche Wege in Graphen
cites work: Über die Maximalzahl fremder unendlicher Wege in Graphen
cites work: Die Maximalzahl fremder zweiseitig unendlicher Wege in Graphen
cites work: A note on Menger's theorem for infinite locally finite graphs
cites work: Über die Maximalzahl kreuzungsfreier H-Wege
cites work: Aspects topologiques de la séparation dans les graphes infinis. I. II
cites work: Développements terminaux des graphes infinis I. Arbres maximaux coterminaux
cites work: Développments terminaux des graphes infinis III. Arbres maximaux sans rayon, cardinalité maximum des ensembles disjoints de rayons
cites work: Q3484856
cites work: A Mengerian theorem for infinite graphs with ideal points
cites work: A minimax theorem for infinite graphs with ideal points
cites work: Improper vertices of a graph
full work available at URL: 0012-365x(92)00525-v
OpenAlex ID: W2089900484
Recommended article: A minimax theorem for infinite graphs with ideal points
Recommended article: A Mengerian theorem for infinite graphs with ideal points
Recommended article: Minimization Problems for Infinite n-Connected Graphs
Recommended article: Q4347900
Recommended article: Q5721213
Recommended article: A bound for minimal graphs with a normal at infinity
Recommended article: Menger's theorem for infinite graphs with ends
Recommended article: Extremal infinite graph theory
Recommended article: Minimizer graphs for a class of extremal problems
Recommended article: Ends and Vertices of Small Degree in Infinite Minimally k-(Edge)-Connected Graphs
description: scientific article; zbMATH DE number 626314