An obstacle to a decomposition theorem for near-regular matroids

TL;DR

This paper demonstrates a fundamental obstacle in extending Seymour's decomposition theorem to near-regular matroids, showing that 3-sums are insufficient for such a decomposition under certain conditions.

Contribution

It identifies a key limitation in decomposing near-regular matroids, proving that 3-sums alone cannot achieve a complete decomposition when basic classes have specific properties.

Findings

3-sums are not sufficient for near-regular matroid decomposition under certain class conditions

A fundamental obstacle in extending Seymour's theorem to near-regular matroids

Basic classes cannot simultaneously contain all graphic and cographic matroids

Abstract

Seymour's Decomposition Theorem for regular matroids states that any matroid representable over both GF(2) and GF(3) can be obtained from matroids that are graphic, cographic, or isomorphic to R10 by 1-, 2-, and 3-sums. It is hoped that similar characterizations hold for other classes of matroids, notably for the class of near-regular matroids. Suppose that all near-regular matroids can be obtained from matroids that belong to a few basic classes through k-sums. Also suppose that these basic classes are such that, whenever a class contains all graphic matroids, it does not contain all cographic matroids. We show that in that case 3-sums will not suffice.