Towards a splitter theorem for internally 4-connected binary matroids VIII: small matroids

TL;DR

This paper uses computer-assisted search to identify all pairs of small internally 4-connected binary matroids satisfying a specific splitter theorem condition, complementing previous theoretical analysis for larger matroids.

Contribution

It provides a complete classification of small matroid pairs satisfying the splitter theorem, filling a gap in the understanding of such structures for matroids with fewer than 16 elements.

Findings

All pairs with |E(M)|<16 satisfying the splitter theorem condition are identified.

The results support the theoretical framework for larger matroids by confirming small cases.

The exhaustive search methodology ensures completeness for small matroids.

Abstract

Our splitter theorem for internally 4-connected binary matroids studies pairs of the form (M,N), where N and M are internally 4-connected binary matroids, M has a proper N-minor, and if M' is an internally 4-connected matroid such that M has a proper M'-minor and M' has an N-minor, then |E(M)|-|E(M')|>3. The analysis in the splitter theorem requires the constraint that |E(M)|>15. In this article, we complement that analysis by using an exhaustive computer search to find all such pairs satisfying |E(M)|<16.