Towards a Splitter Theorem for Internally 4-connected Binary Matroids Iii
نویسنده
چکیده
This paper proves a preliminary step towards a splitter theorem for internally 4-connected binary matroids. In particular, we show that, provided M or M∗ is not a cubic Möbius or planar ladder or a certain coextension thereof, an internally 4-connected binary matroid M with an internally 4-connected proper minor N either has a proper internally 4-connected minor M ′ with an N -minor such that |E(M) − E(M ′)| ≤ 3 or has, up to duality, a triangle T and an element e of T such that M\e has an N -minor and has the property that one side of every 3-separation is a fan with at most four
منابع مشابه
Towards a splitter theorem for internally 4-connected binary matroids II
Let M and N be internally 4-connected binary matroids such that M has a proper N -minor, and |E(N)| ≥ 7. As part of our project to develop a splitter theorem for internally 4-connected binary matroids, we prove the following result: if M\e has no N -minor whenever e is in a triangle of M , and M/e has no N -minor whenever e is in a triad of M , then M has a minor, M ′, such that M ′ is internal...
متن کاملTowards a Splitter Theorem for Internally 4-connected Binary Matroids Viii: Small Matroids
Our splitter theorem studies pairs of the form (M,N), where M and N 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 |E(M)| ≥ 16. In this article, we complement that analysis by des...
متن کاملTowards a splitter theorem for internally 4-connected binary matroids
In our quest to find a splitter theorem for internally 4-connected binary matroids, we proved in the preceding paper in this series that, except when M or its dual is a cubic Möbius or planar ladder or a certain coextension thereof, an internally 4-connected binary matroid M with an internally 4connected proper minor N either has a proper internally 4-connected minor M ′ with an N -minor such t...
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متن کاملTowards a Splitter Theorem for Internally 4-connected Binary Matroids Vii
Let M be a 3-connected binary matroid; M is internally 4connected if one side of every 3-separation is a triangle or a triad, and M is (4, 4, S)-connected if one side of every 3-separation is a triangle, a triad, or a 4-element fan. Assume M is internally 4-connected and that neither M nor its dual is a cubic Möbius or planar ladder or a certain coextension thereof. Let N be an internally 4-con...
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