This is an introductory paper about the category of regular oriented matroids (ROMs). We compare the homotopy types of the categories of regular and binary matroids. For example, in the unoriented case, they have the same fundamental group but we show that the higher homotopy groups are different for rank three regular and binary matroids. We also speculate on the possible impact of a recent theorem of Galatius [Gal] computing the stable cohomology of the category of graphs and on possible applications to higher Reidemeister torsion.
New families of matroids are constructed in this note. These new fami- lies are derived from the concept of linearly independent set family (LISF) introduced by Eicker and Ewald [Linear Algebra and its Applications 388 (2004) 173-191]. The proposed construction generalizes in a natural way the well known class of vectorial matroids over a field.
The classical Whitney's 2-Isomorphism Theorem describes the families of graphs having the same cycle matroid. In this paper we describe the families of graphs having the same truncated cycle matroid and prove, in particular, that every 3-connected graph, except for K4, is uniquely defined by its truncated cycle matroid.
We present two characterizations of regular matroids among orientable matroids and use them to give a measure of "how far" an orientable matroid is from being regular.
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].
Seymour's Decomposition Theorem for regular matroids states that any matroid representable over both GF(2) and GF(3) can be obtained from matroids that are graphic, cographic, or isomorphic to R10 by 1-, 2-, and 3-sums. It is hoped that similar characterizations hold for other classes of matroids, notably for the class of near-regular matroids. Suppose that all near-regular matroids can be obtained from matroids that belong to a few basic classes through k-sums. Also suppose that these basic classes are such that, whenever a class contains all graphic matroids, it does not contain all cographi…
Raghunathan at al. [9] introduced splitting operation with respect to a pair of element for binary matroid and characterized Eulerian binary matroids using it. In general, the splitting operation does not preserve the graphicness property of the given matroids. Shikare and Waphare [10] obtained the characterization for the class of graphic matroids which yield graphic matroids under the splitting operation with respect to a pair of elements. We study the effect of the splitting operation on regular matroids and characterize the class of regular matroids which yield graphic matroids under the s…
A catalogue of all non-isomorphic simple connected regular matroids ${\cal M}$ of cardinality $n \leq 15$ is provided on the net. These matroids are given as binary matrix matroids and are sieved from the large pool of all non-isomorphic binary matrix matroids of cardinality $\leq 15$. For each ${\cal M}$ its Tutte polynomial is determined by an algorithm based on internal and external base activity.
We prove that the non-regular binary matroids with no $P_9^*$-minor have linear growth rate and the maximum size binary matroids with no $P_9^*$-minor are graphic. The main technique in the proof is the Strong Splitter Theorem using which we find the precise infinite families of 3-connected binary matroids with no $P_9^*$-minor.
Let $G$ be a finite undirected multigraph with no self-loops. The Jacobian $\operatorname{Jac}(G)$ is a finite abelian group associated with $G$ whose cardinality is equal to the number of spanning trees of $G$. There are only a finite number of biconnected graphs $G$ such that the exponent of $\operatorname{Jac}(G)$ equals $2$ or $3$. The definition of a Jacobian can also be extended to regular matroids as a generalization of graphs. We prove that there are finitely many connected regular matroids $M$ such that $\operatorname{Jac}(M)$ has exponent $2$ and characterize all such matroids.
We consider decompositions of topes of the oriented matroid realizable as the arrangement of coordinate hyperplanes in $\mathbb{R}^{2^t}$, with respect to a distinguished symmetric $2\cdot 2^t$-cycle in its hypercube graph of topes $\boldsymbol{H}(2^t,2)$. We seek interpretations of such decompositions in the context of subset families on the ground set $E_t:=\{1,\ldots,t\}$ and of the families of their blocking sets, in the context of clutters on $E_t$ and of their blockers.
In unpublished work, Geelen proved that a matroid is near-regular if and only if it has no minor isomorphic to: U2,5; U3,5; the Fano plane and its dual; the non-Fano and its dual; the single-element deletion of AG(2,3), its dual, and the matroid obtained from it with a Delta-Y operation; and P8. We provide a proof of this characterization.
We prove that the extension complexity of the independence polytope of every regular matroid on $n$ elements is $O(n^6)$. Past results of Wong and Martin on extended formulations of the spanning tree polytope of a graph imply a $O(n^2)$ bound for the special case of (co)graphic matroids. However, the case of a general regular matroid was open, despite recent attempts. We also consider the extension complexity of circuit dominants of regular matroids, for which we give a $O(n^2)$ bound.
In the matroid secretary problem we are given a stream of elements and asked to choose a set of elements that maximizes the total value of the set, subject to being an independent set of a matroid given in advance. The difficulty comes from the assumption that decisions are irrevocable: if we choose to accept an element when it is presented by the stream then we can never get rid of it, and if we choose not to accept it then we cannot later add it. Babaioff, Immorlica, and Kleinberg [SODA 2007] introduced this problem, gave O(1)-competitive algorithms for certain classes of matroids, and conje…
We investigate the computational difficulty of approximating the partition function of the ferromagnetic Ising model on a regular matroid. Jerrum and Sinclair have shown that there is a fully polynomial randomised approximation scheme (FPRAS) for the class of graphic matroids. On the other hand, the authors have previously shown, subject to a complexity-theoretic assumption, that there is no FPRAS for the class of binary matroids, which is a proper superset of the class of graphic matroids. In order to map out the region where approximation is feasible, we focus on the class of regular matroid…
In this note we prove a lower bound for the rank of 2-dimensional generic rigidity matroid for regular graphs of degree four and five. Also, we give examples to show the order of the bound we give is sharp.
We present a simple proof of the fact that the base (and independence) polytope of a rank $n$ regular matroid over $m$ elements has an extension complexity $O(mn)$.
Subject to announced results by Geelen, Gerards, and Whittle, we completely characterize the highly connected members of the classes of dyadic, near-regular, and sixth-root-of-unity matroids.
Following the work of Gao and Xie in [2], we state some properties of the inverse Kazhdan-Lusztig polynomial of a matroid. We also give partial answers to a conjecture that states that regular connected matroids are non-degenerate. We link the degeneracy of a matroid to the inverse Kazhdan-Lusztig polynomial and we show that the Conjecture holds for modular matroids, by proving that degenerate modular matroids are not regular.
The first author introduced the circuit-cocircuit reversal system of an oriented matroid, and showed that when the underlying matroid is regular, the cardinalities of such system and its variations are equal to special evaluations of the Tutte polynomial (e.g., the total number of circuit-cocircuit reversal classes equals $t(M;1,1)$, the number of bases of the matroid). By relating these classes to activity classes studied by the first author and Las Vergnas, we give an alternative proof of the above results and a proof of the converse statements that these equalities fail whenever the underly…
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 connectio…
Given a 4-regular graph $F$, we introduce a binary matroid $M_τ(F)$ on the set of transitions of $F$. Parametrized versions of the Tutte polynomial of $M_τ(F)$ yield several well-known graph and knot polynomials, including the Martin polynomial, the homflypt polynomial, the Kauffman polynomial and the Bollobás-Riordan polynomial.
A matroid is $\text{GF}(q)$-regular if it is representable over all proper superfields of the field $\text{GF}(q)$. We show that, for highly connected matroids having a large projective geometry over $\text{GF}(q)$ as a minor, the property of $\text{GF}(q)$-regularity is equivalent to representability over both $\text{GF}(q^2)$ and $\text{GF}(q^t)$ for some odd integer $t \geq 3$. We do this by means of an exact structural description of all such matroids.
In this paper we prove two main results about obstruction to graph planarity. One is that, if $G$ is a 3-connected graph with a $K_5$-minor and $T$ is a triangle of $G$, then $G$ has a $K_5$-minor $H$, such that $E(T)\cont E(H)$. Other is that if $G$ is a 3-connected simple non-planar graph not isomorphic to $K_5$ and $e,f\in E(G)$, then $G$ has a minor $H$ such that $e,f\in E(H)$ and, up to isomorphisms, $H$ is one of the four non-isomorphic simple graphs obtained from $K_{3,3}$ by the addiction of \,0, 1 or 2 edges. We generalize this second result to the class of the regular matroids.
We show that for any regular matroid on $m$ elements and any $α\geq 1$, the number of $α$-minimum circuits, or circuits whose size is at most an $α$-multiple of the minimum size of a circuit in the matroid is bounded by $m^{O(α^2)}$. This generalizes a result of Karger for the number of $α$-minimum cuts in a graph. As a consequence, we obtain similar bounds on the number of $α$-shortest vectors in "totally unimodular" lattices and on the number of $α$-minimum weight codewords in "regular" codes.
The Splitter Theorem states that, if $N$ is a 3-connected proper minor of a 3-connected matroid $M$ such that, if $N$ is a wheel or whirl then $M$ has no larger wheel or whirl, respectively, then there is a sequence $M_0,..., M_n$ of 3-connected matroids with $M_0\cong N$, $M_n=M$ and for $i\in \{1,..., n\}$, $M_i$ is a single-element extension or coextension of $M_{i-1}$. Observe that there is no condition on how many extensions may occur before a coextension must occur. In this paper, we give a strengthening of the Splitter Theorem, as a result of which we can obtain, up to isomorphism, $M$ …
In this paper, we give a complete characterization of binary matroids with no $P_9$-minor. A 3-connected binary matroid $M$ has no $P_9$-minor if and only if $M$ is one of the internally 4-connected non-regular minors of a special 16-element matroid $Y_{16}$, a 3-connected regular matroid, a binary spike with rank at least four, or a matroid obtained by 3-summing copies of the Fano matroid to a 3-connected cographic matroid $M^*(K_{3, n})$, $M^*(K_{3, n}^{\prime})$, $M^*(K_{3, n}^{\prime\prime})$, or $M^*(K_{3, n}^{\prime\prime\prime})$ ($n\ge 2$). Here the simple graphs $K_{3, n}^{\prime}, K_…
Seymour's decomposition theorem for regular matroids is a fundamental result with a number of combinatorial and algorithmic applications. In this work we demonstrate how this theorem can be used in the design of parameterized algorithms on regular matroids. We consider the problem of covering a set of vectors of a given finite dimensional linear space (vector space) by a subspace generated by a set of vectors of minimum size. Specifically, in the Space Cover problem, we are given a matrix M and a subset of its columns T; the task is to find a minimum set F of columns of M disjoint with T such …
The first problem we investigate is the following: given $k\in \mathbb{R}_{\ge 0}$ and a vector $v$ of Plücker coordinates of a point in the real Grassmannian, is the vector obtained by taking the $k$th power of each entry of $v$ again a vector of Plücker coordinates? For $k\neq 1$, this is true if and only if the corresponding matroid is regular. Similar results hold over other fields. We also describe the subvariety of the Grassmannian that consists of all the points that define a regular matroid. The second topic is a related problem for arithmetic matroids. Let $\mathcal{A} = (E, rk, m)$ b…
A multivariate polynomial is stable if it is non-vanishing whenever all variables have positive imaginary parts. A matroid has the weak half-plane property (WHPP) if there exists a stable polynomial with support equal to the set of bases of the matroid. If the polynomial can be chosen with all of its nonzero coefficients equal to one then the matroid has the half-plane property (HPP). We describe a systematic method that allows us to reduce the WHPP to the HPP for large families of matroids. This method makes use of the Tutte group of a matroid. We prove that no projective geometry has the WHP…
In this paper we investigate a family of matroids introduced by Ardila and Billey to study one-dimensional intersections of complete flag arrangements of $\mathbb{C}^n$. The set of lattice points $P_n$ inside the equilateral triangle $S_n$ obtained by intersecting the nonnegative cone of $\mathbb{R}^3$ with the affine hyperplane $x_1 + x_2 + x_3 = n-1$ is the ground set of a matroid $\mathcal{T}_n$ whose independent sets are the subsets $S$ of $P_n$ satisfying that $|S \cap P| \le k$ for each translation $P$ of the set $P_k$. Here we study the structure of the matroids $\mathcal{T}_n$ in conne…
