New minor minimal non-apex graphs

TL;DR

This paper identifies forbidden minors for small apex graphs, lists specific cases, and proves Jørgensen's conjecture for graphs with 13 vertices and minimal degree 6.

Contribution

It provides a comprehensive list of forbidden minors for apex graphs up to certain sizes and confirms Jørgensen's conjecture for a specific class of graphs.

Findings

Forbidden minors for apex graphs with ≤12 vertices identified

Forbidden minors for apex graphs with ≤26 edges listed

Jørgensen's conjecture proven for graphs with 13 vertices and minimal degree 6

Abstract

A graph is apex if it becomes planar after the deletion of one vertex. The family of apex graphs is closed under taking minors, so it is characterized by a finite set of forbidden minors. Determining the finite set of forbidden minors for apex graphs remains an open question. In this paper, we list all forbidden minors for apex graphs with 12 or fewer vertices and all forbidden minors for apex graphs with 26 and fewer edges. We also present graphs outside of these ranges. We show that a graph with 13 vertices and minimal degree 6 is either apex or contains a $K_6$ minor, proving J\o rgensen's conjecture for order 13.