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New minor minimal non-apex graphs

Andrei Pavelescu, Elena Pavelescu, Madeline Potter

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 Jørgensen's conjecture for order 13.

New minor minimal non-apex graphs

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 minor, proving Jørgensen's conjecture for order 13.

Paper Structure

This paper contains 4 sections, 10 theorems, 8 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Graphs with 12 or fewer vertices or 26 or fewer edges
  3. Explanation of computer programs
  4. Graphs of order 13

Key Result

Theorem 1

Let $G$ be a simple graph of order $13$ and minimal degree 6. Then either $G$ contains a $K_6$-minor or it is apex and isomorphic to $K_1\ast Ic$. $\blacktriangleleft$$\blacktriangleleft$

Figures (4)

  • Figure 1: Minor of $G$. The vertices $v_1$ and $v_8$ are adjacent to $a, b, c, d, e,$ and $f$.
  • Figure 2: The neighborhood of $v_1$.
  • Figure 3: Minor of $G$. The vertices $v_1$ and $v_8$ are adjacent to $a, b, c, d, e,$ and $f$.
  • Figure 4: Graphs of order 6 and size 12 that have a vertex of degree 5.

Theorems & Definitions (14)

  • Theorem 1
  • Lemma 2
  • proof
  • Theorem 3: NOPPPa
  • Theorem 4: LPPPa
  • Theorem 5
  • Theorem 6
  • Theorem 7: Mader Ma
  • Theorem 8: Mader Ma
  • Theorem 9: Jørgensen Jor
  • ...and 4 more