A Concise Proof of Discrete Jordan Curve Theorem
  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. 
Li Chen
math.GN