A Concise Proof of Discrete Jordan Curve Theorem

TL;DR

This paper presents a simplified and concise proof of the discrete Jordan curve theorem on discrete surfaces, embedding it in the 2D plane to clarify and make the proof more accessible.

Contribution

It offers a simplified proof of the discrete Jordan curve theorem and clarifies key statements, making the theorem easier to understand and verify.

Findings

Provided a concise proof of the discrete Jordan curve theorem

Embedded the discrete surface in the 2D plane for proof verification

Added appendices to ensure the proof's self-containment and clarity

Abstract

This paper gives a concise proof of the Jordan curve theorem on discrete surfaces. We also embed the discrete surface in the 2D plane to prove the original version of the Jordan curve theorem. This paper is a simple version of L. Chen, Note on the discrete Jordan curve theorem (revised version), arXiv:1312.0316. We seek to clarify and simplify some statements and proofs. Again, the purpose of this paper is to make the proof of the theorems easier to understand. In revision 2, we added Appendix B to make a self-contained proof on verifying simple connectedness of the Euclidean plane in this paper. In this revision, we added a special case for the proof of Theorem 3 in Appendix B that was found when we were revising a new paper for high dimensional contraction. It was easy to resolve in 2D. We put it in Appendix C of this paper.