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

This is the post-print version of the Article - Copyright @ 2012 ElsevierWe prove that if M is a 4-connected binary matroid and N is an internally 4-connected proper minor of M with at least 7 elements, then, unless M is a certain 16-element matroid, there is an element e of E(M) such that either M\e or M/e is internally 4-connected having an N-minor. This strengthens a result of Zhou and is a first step towards obtaining a splitter theorem for internally 4-connected binary matroids.This study is partially funded by Marsden Fund of New Zealand and the National Security Agency

Towards a splitter theorem for internally 4-connected binary matroids VI

Let M be a 3-connected binary matroid; M is called internally 4-connected if one side of every 3-separation is a triangle or a triad, and M is internally 4-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-connected proper minor of M. Our aim is to 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 four elements, or by removing elements in an easily described way from a special substructure of M. When this aim cannot be met, the earlier papers in this series showed that, up to duality, M has a good bowtie, that is, a pair, {x1,x2,x3} and {x4,x5,x6}, of disjoint triangles and a cocircuit, {x2,x3,x4,x5}, where M\x3 has an N-minor and is internally 4-connected. We also showed that, when M has a good bowtie, either M\x3,x6 has an N-minor; or M\x3/x2 has an N-minor and is internally 4-connected. In this paper, we show that, when M\x3,x6 has an N-minor but is not internally 4-connected, M has an internally 4-connected proper minor with an N-minor that can be obtained from M by removing at most three elements, or by removing elements in a well-described way from one of several special substructures of M. This is a significant step towards obtaining a splitter theorem for the class of internally 4-connected binary matroids

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

Splitters and Decomposers for Binary Matroids

Let $EX[M_1\dots, M_k]$ denote the class of binary matroids with no minors isomorphic to $M_1, \dots, M_k$. In this paper we give a decomposition theorem for $EX[S_{10}, S_{10}^*]$, where $S_{10}$ is a certain 10-element rank-4 matroid. As corollaries we obtain decomposition theorems for the classes obtained by excluding the Kuratowski graphs $EX[M(K_{3,3}), M^*(K_{3,3}), M(K_5), M^*(K_5)]$ and $EX[M(K_{3,3}), M^*(K_{3,3})]$. These decomposition theorems imply results on internally $4$-connected matroids by Zhou [\ref{Zhou2004}], Qin and Zhou [\ref{Qin2004}], and Mayhew, Royle and Whitte [\ref{Mayhewsubmitted}].Comment: arXiv admin note: text overlap with arXiv:1403.775

A decomposition theorem for binary matroids with no prism minor

The prism graph is the dual of the complete graph on five vertices with an edge deleted, $K_5\backslash e$. In this paper we determine the class of binary matroids with no prism minor. The motivation for this problem is the 1963 result by Dirac where he identified the simple 3-connected graphs with no minor isomorphic to the prism graph. We prove that besides Dirac's infinite families of graphs and four infinite families of non-regular matroids determined by Oxley, there are only three possibilities for a matroid in this class: it is isomorphic to the dual of the generalized parallel connection of $F_7$ with itself across a triangle with an element of the triangle deleted; it's rank is bounded by 5; or it admits a non-minimal exact 3-separation induced by the 3-separation in $P_9$. Since the prism graph has rank 5, the class has to contain the binary projective geometries of rank 3 and 4, $F_7$ and $PG(3, 2)$, respectively. We show that there is just one rank 5 extremal matroid in the class. It has 17 elements and is an extension of $R_{10}$, the unique splitter for regular matroids. As a corollary, we obtain Dillon, Mayhew, and Royle's result identifying the binary internally 4-connected matroids with no prism minor [5]

Triangle-roundedness in matroids

A matroid $N$ is said to be triangle-rounded in a class of matroids $\mathcal{M}$ if each $3$-connected matroid $M\in \mathcal{M}$ with a triangle $T$ and an $N$-minor has an $N$-minor with $T$ as triangle. Reid gave a result useful to identify such matroids as stated next: suppose that $M$ is a binary $3$-connected matroid with a $3$-connected minor $N$, $T$ is a triangle of $M$ and $e\in T\cap E(N)$; then $M$ has a $3$-connected minor $M'$ with an $N$-minor such that $T$ is a triangle of $M'$ and $|E(M')|\le |E(N)|+2$. We strengthen this result by dropping the condition that such element $e$ exists and proving that there is a $3$-connected minor $M'$ of $M$ with an $N$-minor $N'$ such that $T$ is a triangle of $M'$ and $E(M')-E(N')\subseteq T$. This result is extended to the non-binary case and, as an application, we prove that $M(K_5)$ is triangle-rounded in the class of the regular matroids

Matroids with at least two regular elements

For a matroid $M$, an element $e$ such that both $M\backslash e$ and $M/e$ are regular is called a regular element of $M$. We determine completely the structure of non-regular matroids with at least two regular elements. Besides four small size matroids, all 3-connected matroids in the class can be pieced together from $F_7$ or $S_8$ and a regular matroid using 3-sums. This result takes a step toward solving a problem posed by Paul Seymour: Find all 3-connected non-regular matroids with at least one regular element [5, 14.8.8]