Jonah Blasiak
Describing minimal generating sets of toric ideals is a well-studied and difficult problem. Neil White conjectured in 1980 that the toric ideal associated to a matroid is generated by quadrics corresponding to single element symmetric exchanges. We give a combinatorial proof of White's conjecture for graphic matroids.
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We show that the toric ideal of a lattice path polymatroid is generated by quadrics corresponding to symmetric exchanges, and give a monomial order under which these quadrics form a Gröbner basis. We then obtain an analogous result for lattice path matroids.
The element splitting operation on a graphic matroid, in general may not yield a cographic matroid. In this paper, we give a necessary and sufficient condition for the graphic matroid to yield cographic matroid under the element splitting operation.
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.
Frame matroids and lifted-graphic matroids are two interesting generalizations of graphic matroids. Here we introduce a new generalization, {\em quasi-graphic matroids}, that unifies these two existing classes. Unlike frame matroids and lifted-graphic matroids, it is easy to certify that a matroid is quasi-graphic. The main result of the paper is that every $3$-connected representable quasi-graphic matroid is either a lifted-graphic matroid or a frame matroid.
A frame matroid M is graphic if there is a graph G with cycle matroid isomorphic to M. In general, if there is one such graph, there will be many. Zaslavsky has shown that frame matroids are precisely those having a representation as a biased graph; this class includes graphic matroids, bicircular matroids, and Dowling geometries. Whitney characterized which graphs have isomorphic cycle matroids, and Matthews characterised which graphs have isomorphic graphic bicircular matroids. In this paper, we give a characterization of which biased graphs give rise to isomorphic graphic frame matroids.
Zaslavsky introduced the concept of lifted-graphic matroid. For binary matroids, a binary elementary lift can be defined in terms of the splitting operation. In this paper, we give a method to get a forbidden-minor characterization for the class of graphic matroids whose all lifted-graphic matroids are also graphic using the splitting operation.
We present infinite sequences of excluded minors for both the class of lifted-graphic matroids and the class of frame matroids.
The class of quasi-graphic matroids, recently introduced by Geelen, Gerards, and Whittle, is minor closed and contains both the class of lifted-graphic matroids and the class of frame matroids, each of which generalises the class of graphic matroids. In this paper, we prove that the matroids $U_{3,7}$ and $U_{4,7}$ are the only $9$-connected excluded minors for the class of quasi-graphic matroids.
In this work we provide a decomposition theorem for the class of quaternary and non-binary signed-graphic matroids. This generalizes previous results for binary signed-graphic matroids and graphic matroids, and it provides the theoretical basis for a recognition algorithm.
Frame matroids and lifted-graphic matroids are two distinct minor-closed classes of matroids, each of which generalises the class of graphic matroids. The class of quasi-graphic matroids, recently introduced by Geelen, Gerards, and Whittle, simultaneously generalises both the classes of frame and lifted-graphic matroids. Let $\mathcal{M}$ be one of these three classes, and let $r$ be a positive integer. We show that $\mathcal{M}$ has only a finite number of excluded minors of rank $r$.
Blasiak verified a conjecture of White for graphic matroids by showing that the toric ideal of a graphic matroid is generated by quadrics. In this paper, we extend this result to frame matroids satisfying a linearity condition. Such classes of matroids include graphic matroids, bicircular matroids, signed graphic matroids, and more generally frame matroids obtained from group-labelled graphs.
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 matroid $N$ is a lift of a binary matroid $M$, if $N=Q\backslash X$ when $Q/X=M$ for some binary matroid $Q$ and $X \subseteq E(Q)$ and is called an elementary lift of $M$, if $|X|=1$. A splitting operation on a binary matroid can result in an elementary lift. An elementary lift of a cographic or a graphic matroid need not be cographic or graphic. We intend to characterize the cographic matroids whose elementary lift is a graphic matroid.
In this paper we provide two recognition algorithms for the class of signed-graphic matroids along with necessary and sufficient conditions for a matroid to be signed-graphic. Specifically, we provide a polynomial-time algorithm which determines whether a given binary matroid is signed-graphic and an algorithm which determines whether a general matroid given by an independece oracle is binary signed-graphic.
We prove that there is no polynomial $p(\cdot)$ with the property that a matroid $M$ can be determined to be either a lifted-graphic or frame matroid using at most $p(|M|)$ rank evaluations. This resolves two conjectures of Geelen, Gerards and Whittle (Quasi-graphic matroids, arXiv:1512.03005v1).
Several matroids can be defined on the edge set of a graph. Although historically the cycle matroid has been the most studied, in recent times, the bicircular matroid has cropped up in several places. A theorem of Matthews from late 1970s gives a characterization of graphs whose bicircular matroids are graphic. We give a characterization of graphs whose bicircular matroids are signed-graphic.
An excluded minor characterization for the class of binary signed-graphic matroids with graphic cocircuits is provided. In this report we present the necessary computations for the case analysis in the proof.
An infinite matroid is graphic if all of its finite minors are graphic and the intersection of any circuit with any cocircuit is finite. We show that a matroid is graphic if and only if it can be represented by a graph-like topological space: that is, a graph-like space in the sense of Thomassen and Vella. This extends Tutte's characterization of finite graphic matroids. The representation we construct has many pleasant topological properties. Working in the representing space, we prove that any circuit in a 3-connected graphic matroid is countable.
In this paper, we investigate the importance of column scaling in relating two signed-graphic representations of the same matroid. We used the Sage Mathematics software to generate many examples of signed-graphic matroids and their signed-graphic representations. Our examples show that column scaling is sometimes necessary in order to transform one signed-graphic representation into another; moreover, there exist many collections of signed-graphic representations that row-reduce to the same standard form. We also discuss an interesting matroid-preserving operation on a signed graph, which we c…
Bicircular lift matroids are a class of matroids defined on the edge set of a graph. For a given graph $G$, the circuits of its bicircular lift matroid $L(G)$ are the edge sets of those subgraphs of $G$ that contain at least two cycles, and are minimal with respect to this property. For each cycle $C$ of $G$, since $L(G)/C$ is graphic and most graphic matroids are not bicircular lift, the class of bicircular lift matroids is not minor-closed. In this paper, we prove that the class of matroids that are graphic or bicircular lift has a finite list of excluded minors.
Prohibited Minors For Graphic Matroids That Gives A Binary Gammoid After Splitting
针对拟阵分裂运算无法保持图性、二元gammoid分裂后未必仍为二元gammoid的问题，研究刻画了分裂后可得二元gammoid的图拟阵类，并得到对应图拟阵与余图拟阵的禁用子式。
Cobiased graphs: Single-element extensions and elementary quotients of graphic matroids
本研究提出新的对偶图结构cobiased graph（共偏图），用于刻画图形拟阵的单元素扩张与初等商，是对Zaslavsky偏图相关拟阵刻画工作的对偶性拓展。
In this note we introduce a sufficient condition for the Orlik-Solomon algebra associated to a matroid M to be l-adic and we prove that this condition is necessary when M is binary (in particular graphic). Moreover, this result cannot be extended to the class of all matroids.
The unbreakable quasi-graphic matroids
本文研究了3连通不可破准图型 matroid 的特性，扩展了前人对3连通不可破框架 matroid 的研究成果，并作为特殊情况给出了3连通提升图型 matroid 的特征。
Cographic Splitting Of Graphic Matroids With Respect To A Set With Three Elements
本文研究图拟阵经分裂运算后保持余图性的条件，刻画了关于三元集合分裂得到余图拟阵的图拟阵类别，并为二元情形的已有结论提供了替代证明。
We prove that the maximum size of a simple binary matroid of rank $r \geq 5$ with no AG(3,2)-minor is $\binom{r+1}{2}$ and characterise those matroids achieving this bound. When $r \geq 6$, the graphic matroid $M(K_{r+1})$ is the unique matroid meeting the bound, but there are a handful of smaller examples. In addition, we determine the size function for non-regular simple binary matroids with no AG(3,2)-minor and characterise the matroids of maximum size for each rank.
We give polynomial-time randomized algorithms for computing the girth and the cogirth of binary matroids that are low-rank perturbations of graphic matroids.
In this paper we employ Tutte's theory of bridges to derive a decomposition theorem for binary matroids arising from signed graphs. The proposed decomposition differs from previous decomposition results on matroids that have appeared in the literature in the sense that it is not based on $k$-sums, but rather on the operation of deletion of a cocircuit. Specifically, it is shown that certain minors resulting from the deletion of a cocircuit of a binary matroid will be graphic matroids apart from exactly one that will be signed-graphic, if and only if the matroid is signed-graphic.
The toric variety of a matroid is projectively normal, and therefore it is Cohen-Macaulay. We provide a complete graph-theoretic classification when the toric variety of a graphic matroid is Gorenstein.
A matroid is Gorenstein if its toric variety is. Hibi, Lasoń, Matsuda, Michałek, and Vodička provided a full graph-theoretic classification of Gorenstein matroids associated to simple graphs. We extend this classification to multigraphs.
Let $M$ be a 3-connected binary matroid and let $Y(M)$ be the set of elements of $M$ avoiding at least $r(M)+1$ non-separating cocircuits of $M$. Lemos proved that $M$ is non-graphic if and only if $Y(M)\neq\emp$. We generalize this result when by establishing that $Y(M)$ is very large when $M$ is non-graphic and $M$ has no $M\s(K_{3,3}"')$-minor if $M$ is regular. More precisely that $|E(M)-Y(M)|\le 1$ in this case. We conjecture that when $M$ is a regular matroid with an $M\s(K_{3,3})$-minor, then $r\s_M(E(M)-Y(M))\le 2$. The proof of such conjecture is reduced to a computational verificatio…
Positroid envelopes and graphic positroids
本文通过显式图构造证明每个positroid包络类含图拟阵，并证明图论positroid在包络类中唯一，且可由非负子式符号关联矩阵表示。
$Γ$-graphic delta-matroids and their applications
本文定义了$Γ$-图示delta-拟阵，提出了求解分离问题的多项式时间算法，并利用对称贪婪算法寻找最大权可行集，展示了其在树包装问题中的算法应用。
On the density of matroids omitting a complete-graphic minor
本文证明了排除特定线子式和完全图圈拟阵子式的简单拟阵的密度受单指数函数上界控制，并给出了 frame matroid 特例下接近最优的界。
In 1980, White conjectured that the toric ideal associated to a matroid is generated by binomials corresponding to a symmetric exchange. In this paper, we prove that classes of matroids for which the toric ideal is generated by quadrics and that has quadratic Gröbner bases, are closed under series and parallel extensions, series and parallel connections, and 2-sums.
Real-rootedness of Kazhdan--Lusztig and $Z$-polynomials of thagomizer matroids and graphic matroids of $K_{2,n}$
该研究证明了thagomizer拟阵T_n与K_{2,n}图拟阵的Kazhdan--Lusztig多项式及公共Z-多项式均为实根多项式，通过有理变换将两类多项式的实根性分别归约为切比雪夫多项式零点的交替符号求值和自反多项式的单位圆准则。
Describing Quasi-Graphic Matroids
本文修订并完善了拟图拟阵的完整描述，补充了连通性假设，并提出了一个可在多项式时间内检验第四个公理的替代方案（定理6.20）。
A {\em connectivity function} on a set $E$ is a function $λ:2^E\rightarrow \mathbb R$ such that $λ(\emptyset)=0$, that $λ(X)=λ(E-X)$ for all $X\subseteq E$, and that $λ(X\cap Y)+λ(X\cup Y)\leq λ(X)+λ(Y)$ for all $X,Y \subseteq E$. Graphs, matroids and, more generally, polymatroids have associated connectivity functions. In this paper we give a method for identifying when a connectivity function comes from a graph. This method uses no more than a polynomial number of evaluations of the connectivity function. In contrast, we show that the problem of identifying when a connectivity function comes…
The lattice of integer flows of a graph is known to determine the graph up to 2-isomorphism (work of Su--Wagner and Caporaso--Viviani). In this paper we give an algorithmic construction of the graphic matroid $\calM(G)$ of a graph $G$, given its lattice of integer flows $\calF(G)$. The algorithm can then be applied to compute any other 2-isomorphism invariants (that is, matroid invariants) of $G$ from $\calF(G)$. Our method is based on a result of Amini which describes the relationship between the geometry of the Voronoi cell of $\calF(G)$ and the structure of $G$.
There are only a finite number of excluded minors for the class of bicircular matroids
本文证明了双圈拟阵类仅有有限个排除子式，核心方法是利用偏置图和拟图拟阵表示，证明秩至少为十的排除子式必为拟图拟阵并包含已知小排除子式，从而得出结论。
This paper introduces Dirichlet matroids, a generalization of graphic matroids arising from electrical networks. We present four main theorems. First, we exhibit a matroid quotient involving geometric duals of networks embedded in surfaces with boundary. Second, we characterize the Bergman fans of Dirichlet matroids as subfans of graphic Bergman fans. Third, we prove an interlacing result on the real zeros and poles of the trace of the response matrix. And fourth, we bound the coefficients of the precoloring polynomial of a network by the coefficients of the associated chromatic polynomial.
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…
Myhill-Nerode for hypergraphs and an application to gain-graphic matroids
本文提出超图的Myhill-Nerode定理，定义等价关系并证明其有限指数性，应用于增益图形模组，展示当群Γ非均匀局部有限时，相关模组类无法单模定义。
Tutte polynomials of matroids as universal valuative invariants
本文完整分类了使Tutte多项式成为泛赋值不变量的对偶与子式封闭拟阵族，并由此导出拟阵Tutte多项式的新关系，证明其可唯一表示为图Schubert拟阵Tutte多项式的整系数组合。
Let us consider a truncated matroid $M_Γ^{r}$ of rank $r$ of a graphic matroid of a graph $Γ$. The basis for $M_Γ^{r}$ is the set of the forests with $r$ edges in $Γ$. We consider this basis generating function and compute its Hessian. In this paper, we show that the Hessian of the basis generating function of the truncated matroid of the graphic matroid of the complete or complete bipartite graph does not vanish by calculating the eigenvalues of the Hessian matrix. Moreover, we show that the Hessian matrix of the basis generating function of the truncated matroid of the graphic matroid of the…
Let $M$ be a 3-connected matroid and let $\mathbb F$ be a field. Let $A$ be a matrix over $\mathbb F$ representing $M$ and let $(G,\mathcal B)$ be a biased graph representing $M$. We characterize the relationship between $A$ and $(G,\mathcal B)$, settling four conjectures of Zaslavsky. We show that for each matrix representation $A$ and each biased graph representation $(G,\mathcal{B})$ of $M$, $A$ is projectively equivalent to a canonical matrix representation arising from $G$ as a gain graph over $\mathbb F^+$ or $\mathbb F^\times$ realizing $\mathcal{B}$. Further, we show that the projectiv…
Hypergraphics matroids were studied first by Lorea and later by Frank et al. They can be seen as generalizations of graphic matroids. Here we show that several algorithms developed for the graphic case can be extended to hypergraphic matroids. We treat the following: the separation problem for the associated polytope, testing independence, separation of partition inequalities, computing the rank of a set, computing the strength, computing the arboricity and network reinforcement.
Joins and ear decompositions beyond graphic matroids
本文研究图拟阵之外join最大规模μ(M)与耳分解参数η(M)的等式关系，证明该等式在余图拟阵等类中不成立、相关计算为NP难，同时给出多类拟阵的两参数定量比较界，包括正则拟阵的η(M)≤6μ(M)-2。
Rainbow Stars and Rota's Basis Conjecture for Graphic Matroids
本文研究图拟阵Rota基猜想在星形着色条件下的极端情形，刻画了不同中心、相同中心及双中心星图可分解为彩虹星的条件，并证明将星替换为路时结论不成立。
Failure of the Lefschetz property for the Graphic Matroid
本文针对 Maeno 和 Numata 关于拟阵基生成多项式定义的 Artinian Gorenstein 代数具有强 Lefschetz 性质的猜想，构造了图拟阵反例，并通过构造高阶 Hessian 矩阵核中的非零元素证明其退化性。
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…
The complexity of matroid homomorphism reconfiguration
该研究探讨二元拟阵同态重配置问题${\rm Recol}_\mathbb{M}(N)$的计算复杂性，证明当$N$可拆解为$M(K_2)$时问题平凡，当$N$为$M(K_3)$、$M(K_4)$或含$M(K_5)$的图拟阵时问题为PSPACE完全。
In his pioneering paper on matroids in 1935, Whitney obtained a characterization for binary matroids and left a comment at end of the paper that the problem of characterizing graphic matroids is the same as that of characterizing matroids which correspond to matrices (mod 2) with exactly two ones in each column. Later on Tutte obtained a characterization of graphic matroids in terms of forbidden minors in 1959. It is clear that Whitney indicated about incidence matrices of simple undirected graphs. Here we introduce the concept of a segment binary matroid which corresponds to matrices over $\m…
The excluded 3-minors for vf-safe delta-matroids
本文通过寻找最小障碍即排除3-minors来刻画vf-safe delta-matroids和ribbon-graphic delta-matroids，并解释了binary delta-matroids中3-minors与图vertex minors的等价性。
The Bergman fan of a matroid is the intersection of tropical hyperplanes defined by the circuits. A tropical basis is a subset of the circuits set that defines the Bergman fan. Yu and Yuster posed a question whether every simple regular matroid has a unique minimal tropical basis of its Bergman fan, and verified it for graphic, cographic matroids and $R_{10}$. We show every simple binary matroid has a unique minimal tropical basis. Since the regular matroid is binary, we positively answered the question.
Biconed graphs, weighted forests, and h-vectors of matroid complexes
本文通过双锥图的2-加权森林给出图拟阵h-向量的组合解释，并构造纯多重复形证明Stanley猜想对该类拟阵成立。
A signed circuit cover of a signed graph is a natural analog of a circuit cover of a graph, and is equivalent to a covering of its corresponding signed-graphic matroid with circuits. It was conjectured that a signed graph whose signed-graphic matroid has no coloops has a 6-cover. In this paper, we prove that the conjecture holds for signed Eulerian graphs.
Equivariant Kazhdan--Lusztig Polynomials of Thagomizer Matroids with a Hyperoctahedral Group Action
本文研究了具有超菱形群作用的Thagomizer矩阵的等变Kazhdan-Lusztig多项式，利用回文性与对称群理论计算了多项式及其逆多项式，并通过Frobenius特征表达系数。
Stability conditions and subdivisions of Lawrence polytopes
本文研究图上的稳定性条件与退化集，证明退化集对应图拟阵的单元素扩张，并从 Lawrence 多胞体细分构造稳定性条件，建立与可定向扩张的联系。
We construct minimal cellular resolutions of squarefree monomial ideals arising from hyperplane arrangements, matroids and oriented matroids. These are Stanley-Reisner ideals of complexes of independent sets, and of triangulations of Lawrence matroid polytopes. Our resolution provides a cellular realization of Stanley's formula for their Betti numbers. For unimodular matroids our resolutions are related to hyperplane arrangements on tori, and we recover the resolutions constructed by Bayer, Popescu and Sturmfels. We resolve the combinatorial problems posed in their paper by computing Mobius in…
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