On minors of non-binary matroids

Abstract

No binary matroid has a minor isomorphic toU 24 , the “four-point line”, and Tutte showed that, conversely, every non-binary matroid has aU 24 minor. However, more can be said about the element sets ofU 24 minors and their distribution. Bixby characterized those elements which are inU 24 minors; a matroidM has aU 24 minor using elementx if and only if the connected component ofM containingx is non-binary. We give a similar (but more complicated) characterization for pairs of elements. In particular, we prove that for every two elements of a 3-connected non-binary matroid, there is aU 24 minor using them both.

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Seymour, P.D. On minors of non-binary matroids. Combinatorica 1, 387–394 (1981). https://doi.org/10.1007/BF02579462

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AMS subject classification: (1980)

  • 05 B 35