Vaidy Sivaraman
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Lattice path matroids and bicircular matroids are two well-known classes of transversal matroids. In the seminal work of Bonin and de Mier about structural properties of lattice path matroids, the authors claimed that lattice path matroids significantly differ from bicircular matroids. Recently, it was proved that all cosimple lattice path matroids have positive double circuits, while it was shown that there is a large class of cosimple bicircular matroids with no positive double circuits. These observations support Bonin and de Miers' claim. Finally, Sivaraman and Slilaty suggested studying t…
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.
We characterize the 3-connected members of the intersection of the class of bicircular and cobicircular matroids. Aside from some exceptional matroids with rank and corank at most 5, this class consists of just the free swirls and their minors.
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 are the edge sets of those subgraphs of $G$ that contain at least two cycles, and are minimal with respect to this property. The main result of this paper is a characterization of when two graphs give rise to the same bicircular lift matoid, which answers a question proposed by Irene Pivotto. In particular, aside from some appropriately defined "small" graphs, two graphs have the same bicircular lift matroid if and only if they are $2$-isomorph…
We conjecture that the class of frame matroids can be characterised by a sentence in the monadic second-order logic of matroids, and we prove that there is such a characterisation for the class of bicircular matroids. The proof does not depend on an excluded-minor characterisation.
The excluded minors for the intersection of bicircular and lattice path matroids
本文针对双圈拟阵与格路拟阵的交集类别，确立了其完整的排除子式列表，从而解决了一个最近提出的开放问题。
The first non-trivial case of Hadwiger's conjecture for oriented matroids reads as follows. If $\mathcal{O}$ is an $M(K_4)$-free oriented matroid, then $\mathcal{O}$ admits a NZ $3$-coflow, i.e., it is $3$-colourable in the sense of Hochstättler-Nešetřil. The class of gammoids is a class of $M(K_4)$-free orientable matroids and it is the minimal minor-closed class that contains all transversal matroids. Towards proving the previous statement for the class of gammoids, Goddyn, Hochstättler, and Neudauer conjectured that every gammoid has a positive coline (equivalently, a positive double circui…
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。
We give a fully polynomial-time randomised approximation scheme (FPRAS) for the number of bases in a bicircular matroids. This is a natural class of matroids for which counting bases exactly is #P-hard and yet approximate counting can be done efficiently.
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.
Transversal matroids and the half plane property
本文验证了格路拟阵满足半平面性质，并给出非正阳双圆拟阵示例，证明稀疏铺砌拟阵不满足瑞利性质，为拟阵基负相关猜想提供新视角。
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.
In 30's Hassler Whitney considered and completely solved the problem $(WP)$ of describing the classes of graphs $G$ having the same cycle matroid $M(G)$. A natural analog $(WP)'$ of Whitney's problem $(WP)$ is to describe the classes of graphs $G$ having the same matroid $M'(G)$, where $M'(G)$ is a matroid on the edge set of $G$ distinct from $M(G)$. For example, the corresponding problem $(WP)' = (WP)_{θ}$ for the so-called bicircular matroid $M_{θ}(G)$ of graph $G$ was solved by Coulard, Del Greco and Wagner. In our previous paper [arXive:1508.05364] we introduced and studied the so-called $…
Motivated by work in graph theory, we define the fixing number for a matroid. We give upper and lower bounds for fixing numbers for a general matroid in terms of the size and maximum orbit size (under the action of the matroid automorphism group). We prove the fixing numbers for the cycle matroid and bicircular matroid associated with 3-connected graphs are identical. Many of these results have interpretations through permutation groups, and we make this connection explicit.
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.
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 $r$.
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).
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…
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.
In 30's Hassler Whitney considered and completely solved the problem $(WP)$ of describing the classes of graphs $G$ having the same cycle matroid $M(G)$. A natural analog $(WP)'$ of Whitney's problem $(WP)$ is to describe the classes of graphs $G$ having the same matroid $M'(G)$, where $M'(G)$ is a matroid (on the edge set of $G$) distinct from $M(G)$. For example, the corresponding problem $(WP)'= (WP)_{θ}$ for the so-called bicircular matroid $M_{θ}(G)$ of graph $G$ was solved by Coulard, Del Greco and Wagner. We define the so-called {\em $k$-circular matroid} $M_k(G)$ on the edge set of gra…
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。
Hlineny's Theorem shows that any sentence in the monadic second-order logic of matroids can be tested in polynomial time, when the input is limited to a class of F-representable matroids with bounded branch-width (where F is a finite field). If each matroid in a class can be decomposed by a subcubic tree in such a way that only a bounded amount of information flows across displayed separations, then the class has bounded decomposition-width. We introduce the pigeonhole property for classes of matroids: if every subclass with bounded branch-width also has bounded decomposition-width, then the c…
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。
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…
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…
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。
We give polynomial-time randomized algorithms for computing the girth and the cogirth of binary matroids that are low-rank perturbations of graphic matroids.
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…
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。
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.
We show that almost every matroid contains the rank-3 whirl $\mathcal{W}^3$ or the complete-graphic matroid $M(K_4)$ as a minor.
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.
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.
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。
Oriented Matroid Circuit Polytopes
研究由定向拟阵的带符号回路等产生的多面体，给出图形定向拟阵等相关多面体的维度，明确 A 型相关多面体的面结构、埃尔哈特级数，研究对称群对其作用，揭示这些多面体的多种性质及关联。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。