A proof of Jordan curve theorem based on the sweepline algorithm for trapezoidal decomposition of a polygon

TL;DR

This paper presents the first algorithmic proof of the Jordan curve theorem using a sweepline algorithm in computational geometry, making the proof more accessible for educational purposes.

Contribution

It introduces an algorithmic proof of the Jordan curve theorem based on computational geometry and the sweepline algorithm, utilizing Zorn's lemma.

Findings

Provides the first algorithmic proof of the Jordan curve theorem.

The proof can be incorporated into undergraduate discrete mathematics courses.

Uses the axiom of choice in the proof.

Abstract

We prove the Jordan curve theorem by generalizing the sweepline algorithm for trapezoidal decomposition of a polygon. Our proof uses Zorn's lemma (or, equivalently the axiom of choice). Though several proofs have been given for the Jordan curve theorem by various authors, ours is the {\bf first algorithmic proof} of Jordan curve theorem using computational geometry. Further, with some preparation, the proof can be taught as part of an undergraduate discrete mathematics course, where till now the proof of this theorem was considered inaccessible.