Digital Jordan curves---a graph-theoretical approach to a topological theorem
type: Item
instance of: scholarly article
title: Digital Jordan curves---a graph-theoretical approach to a topological theorem
zbMATH Open document ID: 0778.05050
DOI: 10.1016/0166-8641(92)90017-T
author: Richard G. Wilson
author: V. Neumann-Lara
published in: Topology and its Applications
publication date: 1993-04-01T00:00:00Z
review text: A (graph-theoretical) Jordan curve in a graph is an induced cycle of length greater than 3. A graph is locally Hamiltonian if the neighborhood of every vertex induces a Hamiltonian subgraph. The main theorem of the paper is on locally Hamiltonian, connected, planar graphs \(G\), stating that \(G-C\) contains exactly two connected components for every Jordan curve \(C\) in \(G\). By applying this result to certain infinite planar graphs, as the hexagonal net, or the graph obtained by adding exactly one diagonal in each unit square of the rectangular integer lattice, some digital Jordan curve theorems from \textit{E. Khalimsky}, \textit{R. Kopperman} and \textit{P. R. Meyer} [Topology Appl. 36, No. 1, 1-17 (1990; Zbl 0709.54017)] and Klette and Voss follow immediately.
Mathematics Subject Classification ID: 05C38
Mathematics Subject Classification ID: 54D05
Mathematics Subject Classification ID: 05C45
Mathematics Subject Classification ID: 05C10
zbMATH DE Number: 147516
zbMATH Keywords: locally Hamiltonian graph
zbMATH Keywords: Khalimsky topology
zbMATH Keywords: Jordan curve
zbMATH Keywords: induced cycle
zbMATH Keywords: planar graphs
zbMATH Keywords: hexagonal net
reviewed by: Erich Prisner
MaRDI profile type: Publication
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full work available at URL: 0166-8641(92)90017-t
OpenAlex ID: W1999779131
description: scientific article; zbMATH DE number 147516