Embedding a graph-like continuum in a surface
type: Item
instance of: scholarly article
title: Embedding a graph-like continuum in a surface
zbMATH Open document ID: 1374.54037
author: R. B. Richter
author: Brendan Rooney
published in: Combinatorica
publication date: 2017-03-31T00:00:00Z
review text: A topological space \(X\) is a graph-like space if there is a 0-dimensional subspace \(V\) (these are the vertices of \(X\)) of \(X\) so that \(X\setminus V\) is the disjoint union of connected open sets (these are the edges of \(X\)), each of which is a homeomorph of the real line. In a natural way graph-like continua generalize finite graphs to infinite, compact structures. In the present paper the authors investigate embeddings of graph-like continua in compact surfaces. Generalizing results for embeddings of finite graphs they establish the following facts: (i) every edge is in an open disc that meets the graph-like continuum precisely in that edge; (ii) there are natural analogues of face boundary walks; (iii) there is a graph-like continuum triangulating the same surface and containing as a sub-graph-like continuum the original embedded graph-like continuum; (iv) the face boundaries generate a subspace of the cycle space; (v) the quotient of the cycle space by the boundary cycles is the homology of the surface.
Mathematics Subject Classification ID: 54F50
Mathematics Subject Classification ID: 05C10
Mathematics Subject Classification ID: 57M15
Mathematics Subject Classification ID: 57N05
zbMATH DE Number: 6699166
zbMATH Keywords: graph-like continuum
zbMATH Keywords: surface
zbMATH Keywords: embedding
zbMATH Keywords: face boundary
zbMATH Keywords: cycle space
zbMATH Keywords: \({\mathbb Z}_2\)-homology
zbMATH Keywords: dual graph
reviewed by: Hans Peter Künzi
MaRDI profile type: Publication
full work available at URL: s00493-014-3079-2
OpenAlex ID: W2105506385
cites work: The cycle space of a 3-connected locally finite graph is generated by its finite and infinite peripheral circuits
cites work: MacLane's planarity criterion for locally finite graphs
cites work: Q4236278
cites work: The planarity theorems of MacLane and Whitney for graph-like continua
cites work: The Cycle Space of an Infinite Graph
cites work: Topological paths, cycles and spanning trees in infinite graphs
cites work: Embedding graphs in surfaces
cites work: Q4004123
cites work: On the Classification of Noncompact Surfaces
cites work: The cycle space of an embedded graph
cites work: 3-connected planar spaces uniquely embed in the sphere
cites work: The Jordan-Schonflies Theorem and the Classification of Surface
cites work: Graph-like continua, augmenting arcs, and Menger's theorem
cites work: Cycle spaces in topological spaces
DOI: 10.1007/S00493-014-3079-2
description: scientific article; zbMATH DE number 6699166