Growth rate of binary matroids with no $P_9^*$-minor

TL;DR

This paper establishes that binary matroids excluding a specific minor grow linearly in size and that the largest such matroids are graphic, using the Strong Splitter Theorem to classify their structure.

Contribution

It proves linear growth and characterizes maximum size binary matroids without a $P_9^*$-minor, identifying them as graphic and classifying their structure.

Findings

Binary matroids with no $P_9^*$-minor grow linearly.

Maximum size binary matroids without $P_9^*$-minor are graphic.

Uses Strong Splitter Theorem to classify 3-connected binary matroids.

Abstract

We prove that the non-regular binary matroids with no $P_9^*$-minor have linear growth rate and the maximum size binary matroids with no $P_9^*$-minor are graphic. The main technique in the proof is the Strong Splitter Theorem using which we find the precise infinite families of 3-connected binary matroids with no $P_9^*$-minor.