On Seymour’s Decomposition Theorem

Abstract

The Splitter Theorem states that, if $N$ is a 3-connected proper minor of a 3-connected matroid $M$ such that, if $N$ is a wheel or whirl then $M$ has no larger wheel or whirl, respectively, then there is a sequence $M_0,..., M_n$ of 3-connected matroids with $M_0\cong N$, $M_n=M$ and for $i\in \{1,..., n\}$, $M_i$ is a single-element extension or coextension of $M_{i-1}$. Observe that there is no condition on how many extensions may occur before a coextension must occur. In this paper, we give a strengthening of the Splitter Theorem, as a result of which we can obtain, up to isomorphism, $M$ starting with $N$ and at each step doing a 3-connected single-element extension or coextension, such that at most two consecutive single-element extensions occur in the sequence (unless the rank of the matroids involved are $r(M)$). Moreover, if two consecutive single-element extensions by elements $\{e, f\}$ are followed by a coextension by element $g$, then $\{e, f, g\}$ form a triad in the resulting matroid. Using the Strong Splitter Theorem, we make progress toward the problem of determining the almost-regular matroids [6, 15.9.8]. {\it Find all 3-connected non-regular matroids such that, for all $e$, either $M\backslash e$ or $M/e$ is regular.} In [4] we determined the binary almost-regular matroids with at least one regular element (an element such that both $M\backslash e$ and $M/e$ is regular) by characterizing the class of binary almost-regular matroids with no minor isomorphic to one particular matroid that we called $E_5$. As a consequence of the Strong Splitter Theorem we can determine the class of binary matroids with an $E_5$-minor, but no $E_4$-minor.