A Link Between the Jordan Curve Theorem and the Kuratowski Planarity Criterion

1990 ◽  
Vol 97 (3) ◽  
pp. 216 ◽  
Author(s):  
Carsten Thomassen
Keyword(s):  
Jordan Curve ◽  
Jordan Curve Theorem ◽  
Planarity Criterion

Digital Jordan Curves and Surfaces with Respect to a Closure Operator

2021 ◽  
Vol 179 (1) ◽  
pp. 59-74
Author(s):  
Josef Šlapal
Keyword(s):  
Direct Product ◽  
Jordan Curve ◽  
Closure Operator ◽  
Jordan Curve Theorem ◽  
Closure Operators ◽  
Ternary Relation ◽  
Digital Space ◽  
Jordan Curves ◽  
Curves And Surfaces ◽  
Digital Plane

In this paper, we propose new definitions of digital Jordan curves and digital Jordan surfaces. We start with introducing and studying closure operators on a given set that are associated with n-ary relations (n > 1 an integer) on this set. Discussed are in particular the closure operators associated with certain n-ary relations on the digital line ℤ. Of these relations, we focus on a ternary one equipping the digital plane ℤ2 and the digital space ℤ3 with the closure operator associated with the direct product of two and three, respectively, copies of this ternary relation. The connectedness provided by the closure operator is shown to be suitable for defining digital curves satisfying a digital Jordan curve theorem and digital surfaces satisfying a digital Jordan surface theorem.

scholarly journals A Proof of the Jordan Curve Theorem

2019 ◽  
Vol 11 (12) ◽  
pp. 345-360
Author(s):  
Xing Zhang
Keyword(s):  
Jordan Curve ◽  
Jordan Curve Theorem

Walk-set induced connectedness in digital spaces

2017 ◽  
Vol 33 (2) ◽  
pp. 247-256
Author(s):  
JOSEF SLAPAL ◽  
Keyword(s):  
Jordan Curve ◽  
Digital Images ◽  
Simple Graph ◽  
Jordan Curve Theorem ◽  
Strong Product ◽  
Digital Plane ◽  
Vertex Set ◽  
Product Of Graphs ◽  
Digital Spaces

In an undirected simple graph, we define connectedness induced by a set of walks of the same lengths. We show that the connectedness is preserved by the strong product of graphs with walk sets. This result is used to introduce a graph on the vertex set Z2 with sets of walks that is obtained as the strong product of a pair of copies of a graph on the vertex set Z with certain walk sets. It is proved that each of the walk sets in the graph introduced induces connectedness on Z2 that satisfies a digital analogue of the Jordan curve theorem. It follows that the graph with any of the walk sets provides a convenient structure on the digital plane Z2 for the study of digital images.

The Jordan Curve Theorem

2021 ◽  
pp. 57-76
Author(s):  
James K. Peterson
Keyword(s):  
Jordan Curve ◽  
Jordan Curve Theorem
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