The Kelmans-Seymour conjecture I: special separations

TL;DR

This paper addresses specific types of separations in 5-connected nonplanar graphs to advance the proof of the Kelmans-Seymour conjecture, which posits such graphs contain a subdivision of K_5.

Contribution

It analyzes special 5- and 6-separations, including apex sides, to facilitate the overall proof of the Kelmans-Seymour conjecture.

Findings

Characterization of special 5-separations

Analysis of 6-separations with less restrictive structures

Foundational results for subsequent proof of the conjecture

Abstract

Seymour and, independently, Kelmans conjectured in the 1970s that every 5-connected nonplanar graph contains a subdivision of $K_5$. This conjecture was proved by Ma and Yu for graphs containing $K_4^-$, and an important step in their proof is to deal with a 5-separation in the graph with a planar side. In order to establish the Kelmans-Seymour conjecture for all graphs, we need to consider 5-separations and 6-separations with less restrictive structures. The goal of this paper is to deal with special 5-separations and 6-separations, including those with an apex side. Results will be used in subsequent papers to prove the Kelmans-Seymour conjecture.