Minimum Edge-Outerplanar Embeddings are Polynomial-Time Computable

2026-07-09

AI Summary

This work resolves an open problem posed by Bentz (2009) concerning whether the minimum edge-outerplanarity of a planar graph can be computed in polynomial time. By means of a constructive proof, we present the first polynomial-time algorithm capable of efficiently computing a minimum edge-outerplanar embedding for any planar graph. Our approach integrates techniques from graph theory and algorithm design, complemented by AI-assisted reasoning and rigorous manual verification. This not only establishes the computational tractability of the problem but also offers novel insights and methodological tools for related graph embedding challenges.

Abstract

We prove that the minimum edge-outerplanarity of a planar graph can be computed in polynomial time, resolving an open problem of Bentz (2009). The proof was initially produced by GPT~5.5 Pro and then verified and polished manually.

Problem

• edge-outerplanarity
• planar graph
• polynomial-time computable
• graph embedding

Innovation

• edge-outerplanarity
• polynomial-time algorithm
• planar graph
• graph embedding
• AI-assisted proof