Growth rate of binary matroids with no $P_9^{\ast }$-minor

Abstract

We prove that the non-regular binary matroids with no $P_9^{\ast }$-minor have linear growth rate and the maximum size binary matroids with no $P_9^{\ast }$-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 $P_9^{\ast }$-minor.