Towards a splitter theorem for internally 4-connected binary matroids II

TL;DR

This paper advances the theory of internally 4-connected binary matroids by establishing conditions under which a minor with specific connectivity properties can be obtained with minimal element removal.

Contribution

It proves a new splitter theorem for internally 4-connected binary matroids, identifying conditions that guarantee a minor with the same connectivity and an N-minor with minimal element reduction.

Findings

If certain minors are absent after deletion or contraction in triangles and triads, then a minor with the desired properties exists.

The minor M' has at most two fewer elements than M, maintaining internal 4-connectivity and an N-minor.

The result extends the understanding of the structure of internally 4-connected binary matroids.

Abstract

Let M and N be internally 4-connected binary matroids such that M has a proper N-minor, and |E(N)| is at least seven. As part of our project to develop a splitter theorem for internally 4-connected binary matroids, we prove the following result: if M\e has no N-minor whenever e is in a triangle of M, and M/e has no N-minor whenever e is in a triad of M, then M has a minor, M', such that M' is internally 4-connected with an N-minor, and 0 < |E(M)|-|E(M')| < 3.