Forbidden minors: Finding the finite few

TL;DR

This paper explores the Graph Minor Theorem's implications, demonstrating how finite forbidden minors characterize graph properties, and explicitly determines the seven forbidden minors for the strongly almost-planar property.

Contribution

It provides undergraduate research projects inspired by the Graph Minor Theorem and explicitly identifies forbidden minors for the SAP property.

Findings

Identified the seven forbidden minors for SAP.

Connected the Graph Minor Theorem to practical graph property characterization.

Proposed educational research projects based on forbidden minors.

Abstract

The Graph Minor Theorem of Robertson and Seymour asserts that any graph property, whatsoever, is determined by an associated finite list of graphs. We view this as an impressive generalization of Kuratowski's theorem, which characterizes planarity in terms of two forbidden subgraphs, $K_5$ and $K_{3,3}$. Robertson and Seymour's result empowers students to devise their own Kuratowski type theorems; we propose several undergraduate research projects with that goal. As an explicit example, we determine the seven forbidden minors for a property we call strongly almost--planar (SAP). A graph is SAP if, for any edge $e$, both deletion and contraction of $e$ result in planar graphs.