Sandra R. Kingan

A generalization of a graph result of D. W. Hall

Abstract D.W. Hall proved that every simple 3-connected graph with a K 5 -minor must have a K 3.3 -minor, the only exception being K 5 itself. In this paper, we prove that every 3-connected binary matroid with an M ( K 5 )-minor must have an M ( K 3.3 )- or M *( K 3.3 )-minor, the only exceptions being M ( K 5 ), a highly symmetric 12-element matroid which we call T 12 , and any single-element contraction of T 12 .

Binary matroids without prisms, prism duals, and cubes

Abstract In this paper we determine completely the class of binary matroids with no minors isomorphic to the cycle matroid of the prism graph M * ( K 5 e ), its dual M ( K 5 e ), and the binary affine cube AG (3,2).

A Decomposition Theorem for Binary Matroids with no Prism Minor

The prism graph is the dual of the complete graph on five vertices with an edge deleted, K5\ 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 F7 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 P9. Since the prism graph has rank 5, the class has to contain the binary projective geometries of rank 3 and 4, F7 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 R10, the unique splitter for regular matroids. As a corollary, we obtain Mayhew and Royle’s result identifying the binary internally 4-connected matroids with no prism minor Mayhew and Royle (Siam J Discrete Math 26:755–767, 2012).

On Seymour's Decomposition Theorem

The Splitter Theorem states that, if

On the Circuit-cocircuit Intersection Conjecture

N

Intersections of circuits and cocircuits in binary matroids

is a 3-connected proper minor of a 3-connected matroid

On the matroids in which all hyperplanes are binary

M

Almost-Graphic Matroids

such that, if

Strong Splitter Theorem

N

A NOTE ON GF(5)-REPRESENTABLE MATROIDS

is a wheel or whirl then