9 research outputs found
Towards a splitter theorem for internally 4-connected binary matroids VIII: small matroids
Our splitter theorem for internally 4-connected binary matroids studies pairs
of the form (M,N), where N and M are internally 4-connected binary matroids, M
has a proper N-minor, and if M' is an internally 4-connected matroid such that
M has a proper M'-minor and M' has an N-minor, then |E(M)|-|E(M')|>3. The
analysis in the splitter theorem requires the constraint that |E(M)|>15. In
this article, we complement that analysis by using an exhaustive computer
search to find all such pairs satisfying |E(M)|<16.Comment: Correcting minor error
Towards a splitter theorem for internally 4-connected binary matroids IX: The theorem
Let M be a binary matroid that is internally 4-connected, that is, M is 3-connected, and one side of every 3-separation is a triangle or a triad. Let N be an internally 4-connected proper minor of M. In this paper, we show that M has a proper internally 4-connected minor with an N-minor that can be obtained from M either by removing at most three elements, or by removing some set of elements in an easily described way from one of a small collection of special substructures of M
The excluded minors for the class of matroids that are binary or ternary
We show that the excluded minors for the class of matroids that are binary or ternary are U2,5, U3,5, U2,4position indicatorF7, U2,4position indicatorF7*, U2,4position indicator2F7, U2,4position indicator2F7*, and the unique matroids obtained by relaxing a circuit-hyperplane in either AG(3,2) or T12. The proof makes essential use of results obtained by Truemper on the structure of almost-regular matroids. © 2011 Elsevier Ltd
LIPIcs, Volume 244, ESA 2022, Complete Volume
LIPIcs, Volume 244, ESA 2022, Complete Volum
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum