Forbidden minors: Finding the finite few

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.