Internally $4$-connected binary matroids with every element in three triangles

TL;DR

This paper proves that internally 4-connected binary matroids with each element in three triangles have at least four elements whose contraction preserves internal 4-connectivity.

Contribution

It establishes a new structural property of such binary matroids, identifying elements that maintain connectivity upon contraction.

Findings

At least four elements e exist such that si(M/e) is internally 4-connected.

The result applies to binary matroids with specific triangle configurations.

Provides insights into the structure and connectivity preservation in binary matroids.

Abstract

Let $M$ be an internally $4$-connected binary matroid with every element in three triangles. Then $M$ has at least four elements $e$ such that si$(M/e)$ is internally 4-connected.