Minor-closed classes of binary functions

TL;DR

This paper characterizes minor-closed classes of binary functions, generalizing matroid theory, and provides new proofs for classical results like Tutte's excluded minor theorem within this broader framework.

Contribution

It offers the first comprehensive excluded minor characterizations for classes of binary functions with well-defined minors and rank functions, extending matroid theory.

Findings

Characterization of classes of binary functions with well-defined minors

Excluded minor characterizations for polymatroids, matroids, and binary matroids

New proof of Tutte's excluded minor theorem for binary matroids

Abstract

Binary functions are a generalisation of the cocircuit spaces of binary matroids to arbitrary functions. Every rank function is assigned a binary function, and the deletion and contraction operations of binary functions generalise matroid deletion and contraction. We give the excluded minor characterisations for the classes of binary functions with well defined minors, and those with an associated rank function. Within these classes, we also characterise the classes of binary functions corresponding to polymatroids, matroids and binary matroids by their excluded minors. This gives a new proof of Tutte's excluded minor characterisation of binary matroids in the more generalised space of binary functions.