Characterizing binary matroids with no P9P9-minor

In this paper, we give a complete characterization of binary matroids with no P9P9-minor. A 3-connected binary matroid M   has no P9P9-minor if and only if M   is a 3-connected regular matroid, a binary spike with rank at least four, one of the internally 4-connected non-regular minors of a special 16-element matroid Y16Y16, or a matroid obtained by 3-summing copies of the Fano matroid to a 3-connected cographic matroid M⁎(K3,n)M⁎(K3,n), M⁎(K3,n′), M⁎(K3,n″), or M⁎(K3,n‴) (n≥2n≥2). Here the simple graphs K3,n′, K3,n″, and K3,n‴ are obtained from K3,nK3,n by adding one, two, or three edges in the color class of size three, respectively.