Characterizing binary matroids with no P₉-minor

In this paper, we give a complete characterization of binary matroids with no $P_9$-minor. A 3-connected binary matroid $M$ has no $P_9$-minor if and only if $M$ is one of the internally 4-connected non-regular minors of a special 16-element matroid $Y_{16}$, a 3-connected regular matroid, a binary spike with rank at least four, or a matroid obtained by 3-summing copies of the Fano matroid to a 3-connected cographic matroid $M^*(K_{3, n})$, $M^*(K_{3, n}^{\prime})$, $M^*(K_{3, n}^{\prime\prime})$, or $M^*(K_{3, n}^{\prime\prime\prime})$ ($n\ge 2$). Here the simple graphs $K_{3, n}^{\prime}, K_{3, n}^{\prime\prime}$, and $K_{3, n}^{\prime\prime\prime}$ are obtained from $K_{3, n}$ by adding one, two, or three edges in the color class of size three, respectively.