Cographic Matroids Research Articles

The Jones polynomial from a Goeritz matrix

AbstractWe give an explicit algorithm for calculating the Kauffman bracket of a link diagram from a Goeritz matrix for that link. Further, we show how the Jones polynomial can be recovered from a Goeritz matrix when the corresponding checkerboard surface is orientable, or when more information is known about its Gordon–Litherland form. In the process we develop a theory of Goeritz matrices for cographic matroids, which extends the bracket polynomial to any symmetric integer matrix. We place this work in the context of links in thickened surfaces.

Supereulerian regular matroids without small cocircuits

AbstractA cycle of a matroid is a disjoint union of circuits. A matroid is supereulerian if it contains a spanning cycle. To answer an open problem of Bauer in 1985, Catlin proved in [J. Graph Theory 12 (1988) 29–44] that for sufficiently large , every 2‐edge‐connected simple graph with and minimum degree is supereulerian. In [Eur. J. Combinatorics, 33 (2012), 1765–1776], it is shown that for any connected simple regular matroid , if every cocircuit of satisfies , then is supereulerian. We prove the following. (i) Let be a connected simple regular matroid. If every cocircuit of satisfies , then is supereulerian. (ii) For any real number with , there exists an integer such that if every cocircuit of a connected simple cographic matroid satisfies , then is supereulerian.

Even circuits in oriented matroids

In this paper we generalise the even directed cycle problem, which asks whether a given digraph contains a directed cycle of even length, to orientations of regular matroids. We define non-even oriented matroids generalising non-even digraphs, which played a central role in resolving the computational complexity of the even dicycle problem. Then we show that the problem of detecting an even directed circuit in a regular matroid is polynomially equivalent to the recognition of non-even oriented matroids. Our main result is a precise characterisation of the class of non-even oriented cographic matroids in terms of forbidden minors, which complements an existing characterisation of non-even oriented graphic matroids by Seymour and Thomassen.Mathematics Subject Classifications: 05B35, 05C20, 05C70, 05C75, 05C83, 05C85, 52C40

Characterizing matroids whose bases form graphic delta-matroids

We introduce delta-graphic matroids, which are matroids whose bases form graphic delta-matroids. The class of delta-graphic matroids contains graphic matroids as well as cographic matroids and is a proper subclass of the class of regular matroids. We give a structural characterization of the class of delta-graphic matroids. We also show that every forbidden minor for the class of delta-graphic matroids has at most 48 elements.

On characterizing the class of cographic signed-graphic matroids

Necessary and sufficient conditions for a cographic matroid having no two special matroids as minors to be signed-graphic are provided. The characterization utilizes the notion of Fournier triples of cocircuits and the recent excluded-minor characterization for regular signed-graphic matroids.

Short rainbow cycles in graphs and matroids

AbstractLet be a simple ‐vertex graph and be a coloring of with colors, where each color class has size at least 2. We prove that contains a rainbow cycle of length at most , which is best possible. Our result settles a special case of a strengthening of the Caccetta‐Häggkvist conjecture, due to Aharoni. We also show that the matroid generalization of our main result also holds for cographic matroids, but fails for binary matroids.

Spanning Circuits in Regular Matroids

We consider the fundamental Matroid Theory problem of finding a circuit in a matroid containing a set T of given terminal elements. For graphic matroids, this corresponds to the problem of finding a simple cycle passing through a set of given terminal edges in a graph. The algorithmic study of the problem on regular matroids, a superclass of graphic matroids, was initiated by Gavenčiak, Král’, and Oum [ICALP’12], who proved that the case of the problem with ∣T∣ = 2 is fixed-parameter tractable (FPT) when parameterized by the length of the circuit. We extend the result of Gavenčiak, Král’, and Oum by showing that for regular matroids • the M inimum S panning C ircuit problem, deciding whether there is a circuit with at most ℓ elements containing T , is FPT parameterized by k = ℓ − ∣T∣ • the S panning C ircuit problem, deciding whether there is a circuit containing ∣T∣, is FPT parameterized by ∣T∣. We note that extending our algorithmic findings to binary matroids, a superclass of regular matroids, is highly unlikely: M inimum S panning C ircuit parameterized by ℓ is W[1]-hard on binary matroids even when ∣T∣ = 1. We also show a limit to how far our results can be strengthened by considering a smaller parameter. More precisely, we prove that M inimum S panning C ircuit parameterized by ∣T∣ is W[1]-hard even on cographic matroids, a proper subclass of regular matroids.

Interpretations of the Tutte polynomials of regular matroids

A regular chain group N is the set of integral vectors orthogonal with rows of a matrix representing a regular matroid M, i.e., a totally unimodular matrix. N correspond to the set of flows and tensions if M is a graphic and cographic matroid, respectively. We evaluate the Tutte polynomial of M as number of pairs of specified elements of N.

Editing to Connected F-Degree Graph

In the Edge Editing to Connected $f$-Degree Graph problem we are given a graph $G$, an integer $k$, and a function $f$ assigning integers to vertices of $G$. The task is to decide whether there is a connected graph $F$ on the same vertex set as $G$, such that for every vertex $v$, its degree in $F$ is $f(v)$, and the number of edges in $E(G)\triangle E(F)$, the symmetric difference of $E(G)$ and $E(F)$, is at most $k$. We show that Edge Editing to Connected $f$-Degree Graph is fixed-parameter tractable (FPT) by providing an algorithm solving the problem on an $n$-vertex graph in time $2^{\mathcal O(k)}n^{\mathcal O(1)}$. We complement this result by showing that the weighted version of the problem with costs $1$ and $0$ is W[1]-hard when parameterized by $k$ and the maximum value of $f$ even when the input graph is a tree. Our FPT algorithm is based on a nontrivial combination of color-coding and fast computations of representative families over the direct sum matroid of $\ell$-elongation of the co-graphic matroid associated with $G$ and a uniform matroid over the set of nonedges of $G$. We believe that this combination could be useful in designing parameterized algorithms for other edge editing and connectivity problems.

Schottky algorithms: Classical meets tropical

We present a new perspective on the Schottky problem that links numerical computing with tropical geometry. The task is to decide whether a symmetric matrix defines a Jacobian, and, if so, to compute the curve and its canonical embedding. We offer solutions and their implementations in genus four, both classically and tropically. The locus of cographic matroids arises from tropicalizing the Schottky-Igusa modular form.

Finding even subgraphs even faster

In the Undirected Eulerian Edge Deletion problem, we are given an undirected graph and an integer k, and the objective is to delete k edges such that the resultant graph is a connected graph in which all the vertices have even degrees. The corresponding problem in digraphs where the resulting graph should be strongly connected and every vertex should have the same in-degree as its out-degree is called Directed Eulerian Edge Deletion. In this paper, using the technique of computing representative families of co-graphic matroids we design algorithms which solve these problems in time 2O(k)nO(1), improving the algorithms by Cygan et al. [Algorithmica, 2014] and affirmatively answer the open problem posed by them. The crucial insight we bring to these problems is to view the solution as an independent set of a co-graphic matroid.

How Many Circuits Determine an Oriented Matroid?

Las Vergnas & Hamidoune studied the number of circuits needed to determine an oriented matroid. In this paper we investigate this problem and some new variants, as well as their interpretation in particular classes of matroids. We present general upper and lower bounds in the setting of general connected orientable matroids, leading to the study of subgraphs of the base graph and the intersection graph of circuits. We then consider the problem for uniform matroids which is closely related to the notion of (connected) covering numbers in Design Theory. Finally, we also devote special attention to regular matroids as well as some graphic and cographic matroids leading in particular to the topics of (connected) bond and cycle covers in Graph Theory.

Templates for Binary Matroids

A binary frame template is a device for creating binary matroids from graphic or cographic matroids. Such matroids are said to conform or coconform to the template. We introduce a preorder on these templates and determine the nontrivial templates that are minimal with respect to this order. As an application of our main result, we determine the eventual growth rates of certain minor-closed classes of binary matroids, including the class of binary matroids with no minor isomorphic to PG(3,2). Our main result applies to all highly-connected matroids in a class, not just those of maximum size. As a second application, we characterize the highly-connected 1-flowing matroids.

Single Commodity-Flow Algorithms for Lifts of Graphic and CoGraphic Matroids

Consider a binary matroid $M$ given by its matrix representation. We show that if $M$ is a lift of a graphic or cographic matroid, then in polynomial time (in the size of $M$ and encoding length of the weights) we can either solve the single commodity-flow problem for $M$ or find an obstruction for which the max-flow min-cut relation does not hold. The key tool is an algorithmic version of Lehman's theorem for set covering polyhedra. This tool relies on the ellipsoid method.

Characterizing binary matroids with no [formula omitted]-minor

In this paper, we give a complete characterization of binary matroids with no P9-minor. A 3-connected binary matroid M has no P9-minor if and only if M is a 3-connected regular matroid, a binary spike with rank at least four, one of the internally 4-connected non-regular minors of a special 16-element matroid Y16, or a matroid obtained by 3-summing copies of the Fano matroid to a 3-connected cographic matroid M⁎(K3,n), M⁎(K3,n′), M⁎(K3,n″), or M⁎(K3,n‴) (n≥2). Here the simple graphs K3,n′, K3,n″, and K3,n‴ are obtained from K3,n by adding one, two, or three edges in the color class of size three, respectively.

On Regular Matroids Without Certain Minors

We extend several excluded-minor characterizations of graphs without certain minors to the class of regular matroids. By Seymour’s decomposition of the regular matroids, any regular matroid without a certain minor N can be constructed from N-free graphic matroids, cographic matroids, and \({R_10}\) (if \({R_10}\) is N-free) using 1-, 2-, and 3-sums. The converse, however, is not necessarily true. For example, the \({K_{3, 3}}\)-free graphs \({K_{5}\ e}\) and \({K_4}\) can be 3-summed over a certain triangle to form the graph \({K_{3, 3}}\). In this paper, we first prove a decomposition theorem of regular matroids without a vertically 4-connected minor with rank exceeding three. As a result, we extend to regular matroids some graph excluded minors results of Ding for the octahedron, Maharry for the cube, and Robertson for the Mobius ladder on eight vertices.

Graphic splitting of cographic matroids

In this paper, we obtain a forbidden minor characterization of a cographic matroid M for which the splitting matroid Mx,y is graphic for every pair x, y of elements of M .

A forbidden-minor characterization for the class of regular matroids which yield the cographic es-splitting matroids

Knowing the excluded minors for a minor-closed matroid property provides a useful alternative characterization of that property. In this paper, we find a forbidden-minor characterization for the class of regular matroids which yield the cographic matroids under the es-splitting operation.

Graphs whose flow polynomials have only integral roots

We show if the flow polynomial of a bridgeless graph G has only integral roots, then G is the dual graph to a planar chordal graph. We also show that for 3-connected cubic graphs, the same conclusion holds under the weaker hypothesis that it has only real flow roots. Expressed in the language of matroid theory, this result says that the cographic matroids with only integral characteristic roots are the cycle matroids of planar chordal graphs.

An Obstacle to a Decomposition Theorem for Near-Regular Matroids

Seymour's decomposition theorem [J. Combin. Theory Ser. B, 28 (1980), pp. 305–359] for regular matroids states that any matroid representable over both $\mathrm{GF}(2)$ and $\mathrm{GF}(3)$ can be obtained from matroids that are graphic, cographic, or isomorphic to $R_{10}$ 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 cographic matroids. We show that, in that case, 3-sums will not suffice.