On Seymour's Decomposition Theorem

Abstract

Let $\mathcal{M}$ be a class of matroids closed under minors and isomorphism. Let $N$ be a matroid in $\mathcal{M}$ with an exact $k$-separation $(A,B)$. We say $N$ is a $k$-decomposer for $\mathcal{M}$ having $(A,B)$ as an inducer, if every matroid $M\in \mathcal{M}$ having $N$ as a minor has a $k$-separation $(X,Y)$ such that, $A\subseteq X$ and $B\subseteq Y$. Seymour [3, 9.1] proved that a matroid $N$ is a $k$-decomposer for an excluded-minor class, if certain conditions are met for all 3-connected matroids $M$ in the class, where $|E(M)-E(N)|\le 2$. We reinterpret Seymour's Theorem in terms of the connectivity function and give a check-list that is easier to implement because case-checking is reduced.

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