A decomposition theorem for binary matroids with no prism minor

TL;DR

This paper characterizes binary matroids with no prism minor, revealing their structure, bounds on rank, and identifying a unique extremal matroid, thus extending classical graph minor results to matroid theory.

Contribution

It provides a decomposition theorem for binary matroids excluding the prism minor, identifying their structure and extremal properties, and connects to classical graph minor results.

Findings

Class of binary prism-minor-free matroids characterized

Only three structural possibilities for such matroids

Unique rank 5 extremal matroid with 17 elements identified

Abstract

The prism graph is the dual of the complete graph on five vertices with an edge deleted, $K_5\backslash e$. In this paper we determine the class of binary matroids with no prism minor. The motivation for this problem is the 1963 result by Dirac where he identified the simple 3-connected graphs with no minor isomorphic to the prism graph. We prove that besides Dirac's infinite families of graphs and four infinite families of non-regular matroids determined by Oxley, there are only three possibilities for a matroid in this class: it is isomorphic to the dual of the generalized parallel connection of $F_7$ with itself across a triangle with an element of the triangle deleted; it's rank is bounded by 5; or it admits a non-minimal exact 3-separation induced by the 3-separation in $P_9$. Since the prism graph has rank 5, the class has to contain the binary projective geometries of rank 3 and 4, $F_7$ and $PG(3, 2)$, respectively. We show that there is just one rank 5 extremal matroid in the class. It has 17 elements and is an extension of $R_{10}$, the unique splitter for regular matroids. As a corollary, we obtain Dillon, Mayhew, and Royle's result identifying the binary internally 4-connected matroids with no prism minor [5].