1,196 research outputs found

    Triangles in 3-connected matroids

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    AbstractA collection F of 3-connected matroids is triangle-rounded if, whenever M is a 3-connected matroid having a minor in F, and T is a 3-element circuit of M, then M has a minor which uses T and is isomorphic to a member of F. An efficient theorem for testing a collection of matroids for this property is presented. This test is used to obtain several results including the following extension of a result of Asano, Nishizeki, and Seymour. Let T be a 3-element circuit of a 3-connected binary nonregular matroid M with at least eight elements. Then M has a minor using T that is isomorphic to S8 or the generalized parallel connection across T of F7 and M(K4)

    Fork-decompositions of matroids

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    For the abstract of this paper, please see the PDF file

    On Density-Critical Matroids

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    For a matroid MM having mm rank-one flats, the density d(M)d(M) is mr(M)\tfrac{m}{r(M)} unless m=0m = 0, in which case d(M)=0d(M)= 0. A matroid is density-critical if all of its proper minors of non-zero rank have lower density. By a 1965 theorem of Edmonds, a matroid that is minor-minimal among simple matroids that cannot be covered by kk independent sets is density-critical. It is straightforward to show that U1,k+1U_{1,k+1} is the only minor-minimal loopless matroid with no covering by kk independent sets. We prove that there are exactly ten minor-minimal simple obstructions to a matroid being able to be covered by two independent sets. These ten matroids are precisely the density-critical matroids MM such that d(M)>2d(M) > 2 but d(N)2d(N) \le 2 for all proper minors NN of MM. All density-critical matroids of density less than 22 are series-parallel networks. For k2k \ge 2, although finding all density-critical matroids of density at most kk does not seem straightforward, we do solve this problem for k=94k=\tfrac{9}{4}.Comment: 16 page

    Constructing internally 4-connected binary matroids

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    This is the post-print version of the Article - Copyright @ 2013 ElsevierIn an earlier paper, we proved that an internally 4-connected binary matroid with at least seven elements contains an internally 4-connected proper minor that is at most six elements smaller. We refine this result, by giving detailed descriptions of the operations required to produce the internally 4-connected minor. Each of these operations is top-down, in that it produces a smaller minor from the original. We also describe each as a bottom-up operation, constructing a larger matroid from the original, and we give necessary and su fficient conditions for each of these bottom-up moves to produce an internally 4-connected binary matroid. From this, we derive a constructive method for generating all internally 4-connected binary matroids.This study is supported by NSF IRFP Grant 0967050, the Marsden Fund, and the National Security Agency

    A chain theorem for internally 4-connected binary matroids

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    This is the post-print version of the Article - Copyright @ 2011 ElsevierLet M be a matroid. When M is 3-connected, Tutte’s Wheels-and-Whirls Theorem proves that M has a 3-connected proper minor N with |E(M) − E(N)| = 1 unless M is a wheel or a whirl. This paper establishes a corresponding result for internally 4-connected binary matroids. In particular, we prove that if M is such a matroid, then M has an internally 4-connected proper minor N with |E(M) − E(N)| at most 3 unless M or its dual is the cycle matroid of a planar or Möbius quartic ladder, or a 16-element variant of such a planar ladder.This study was partially supported by the National Security Agency

    Matroids with at least two regular elements

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    For a matroid MM, an element ee such that both M\eM\backslash e and M/eM/e are regular is called a regular element of MM. 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 F7F_7 or S8S_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]

    Triangle-roundedness in matroids

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    A matroid NN is said to be triangle-rounded in a class of matroids M\mathcal{M} if each 33-connected matroid MMM\in \mathcal{M} with a triangle TT and an NN-minor has an NN-minor with TT as triangle. Reid gave a result useful to identify such matroids as stated next: suppose that MM is a binary 33-connected matroid with a 33-connected minor NN, TT is a triangle of MM and eTE(N)e\in T\cap E(N); then MM has a 33-connected minor MM' with an NN-minor such that TT is a triangle of MM' and E(M)E(N)+2|E(M')|\le |E(N)|+2. We strengthen this result by dropping the condition that such element ee exists and proving that there is a 33-connected minor MM' of MM with an NN-minor NN' such that TT is a triangle of MM' and E(M)E(N)TE(M')-E(N')\subseteq T. This result is extended to the non-binary case and, as an application, we prove that M(K5)M(K_5) is triangle-rounded in the class of the regular matroids

    Unavoidable parallel minors of regular matroids

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    This is the post-print version of the Article - Copyright @ 2011 ElsevierWe prove that, for each positive integer k, every sufficiently large 3-connected regular matroid has a parallel minor isomorphic to M (K_{3,k}), M(W_k), M(K_k), the cycle matroid of the graph obtained from K_{2,k} by adding paths through the vertices of each vertex class, or the cycle matroid of the graph obtained from K_{3,k} by adding a complete graph on the vertex class with three vertices.This study is partially supported by a grant from the National Security Agency

    Internally 4-connected binary matroids with cyclically sequential orderings

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    We characterize all internally 4-connected binary matroids M with the property that the ground set of M can be ordered (e0,…,en−1) in such a way that {ei,…,ei+t} is 4-separating for all 0≤i,t≤n−1 (all subscripts are read modulo n). We prove that in this case either n≤7 or, up to duality, M is isomorphic to the polygon matroid of a cubic or quartic planar ladder, the polygon matroid of a cubic or quartic Möbius ladder, a particular single-element extension of a wheel, or a particular single-element extension of the bond matroid of a cubic ladder
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