Computing the Excluded Minors for Branch-Width 3 over Small Fields

by Petr Hlineny

This is an online addition to the paper [Petr Hlineny, On the Excluded Minors for Matroids of Branch-Width Three, submitted 2002], using the MACEK matroid computing package.

Binary Excluded Minors

We provide a computer-assisted proof for the following finite-case statement:

Lemma 4.4 Let M be a 3-connected binary matroid on at most 14 elements with an F7-minor. If M has branch-width 4, then M has an N-minor for some N in F2.

Here F2 = {grQ3, grO6, grK5, grK5*, grV8, grV8*, R10, ND11, ND14, ND23}.

For the proof we use a Macek procedure bw3bin provided for download. This procedure and its use is extensively described here.

Ternary Excluded Minors

We provide a computer-assisted proof for the following finite-case statement:

Proposition 4.5 Let F3 be the set of (pairwise non-isomorphic) excluded minors for branch-width 3 that are ternary but not binary. Then F3 contains no matroids on less than 9 elements, 18 matroids on 9 elements, 31 matroids on 10 elements, and no matroid on 11 or 12 elements.

The set F3 on up to 12 elements is here, and a Macek procedure bw3tern here. This procedure and its use is extensively described here.

Quaternary Excluded Minors

We provide a computer-assisted proof for the following finite-case statement:

Proposition 4.6 Let F4 be the set of (pairwise non-isomorphic) excluded minors for branch-width 3 that are quaternary but neither ternary nor binary. Then F4 contains no matroids on less than 8 elements, 5 matroids on 8 elements, 90 matroids on 9 elements, and 32 matroids on 10 elements.

The set F4 on up to 10 elements is here, and a Macek procedure bw3quat here. This procedure and its use is extensively described here.


Copyright (C) 2001,2002 Petr Hlineny,
Petr.Hlineny!nosp@m!vuw.ac.nz or Petr.Hlineny!nosp@m!seznam.cz

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