On planarity of graphs in 3-manifolds
type: Item
instance of: scholarly article
title: On planarity of graphs in 3-manifolds
zbMATH Open document ID: 0812.57005
DOI: 10.1007/BF02566522
author: Ying-Qing Wu
published in: Commentarii Mathematici Helvetici
publication date: 1993-05-16T00:00:00Z
full work available at URL: https://eudml.org/doc/140288
review text: The author defines a graph \(\Gamma\) in a 3-manifold \(M\) to be planar (I would prefer to say planar in \(M\)) if \(\Gamma\) lies in a 2-sphere in \(M\) and then studies the problem of when an abstractly planar graph in a 3- manifold \(M\) is planar (in \(M\)). One result: \(\Gamma\) is planar if and only if each cycle of \(\Gamma\) bounds a disk in \(M\) whose interior misses \(\Gamma\). This is used to give another proof of a theorem of Scharlemann and Thompson that an abstractly planar graph \(\Gamma\) in \(S^ 3\) is planar if every proper subgraph is planar and \(\pi_ 1(S^ 3 - \Gamma)\) is a free group.
reviewed by: John Hempel
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 05C99
Mathematics Subject Classification ID: 57N10
zbMATH DE Number: 167712
zbMATH Keywords: planar in \(M\)
zbMATH Keywords: planar graph in a 3-manifold
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 167712