Daryl Funk, Dillon Mayhew, Mike Newman
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.
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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…
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.
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…
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。
Monadic transductions and definable classes of matroids
本文系统建立拟阵单子二阶转导理论，证明后向翻译定理，并据此证明格路拟阵类与尖刺子式类可由单子二阶逻辑定义。
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.
A new minor-closed class of transversal matroids
本文刻画了横截拟阵单元素收缩仍为横截拟阵的条件，据此定义了新的次要闭横截拟阵类——路径-循环拟阵，其推广了双循环拟阵和多路径拟阵。
We conjecture that it is not possible to finitely axiomatize matroid representability in monadic second-order logic for matroids, and we describe some partial progress towards this conjecture. We present a collection of sentences in monadic second-order logic and show that it is possible to finitely axiomatize matroids using only sentences in this collection. Moreover, we can also axiomatize representability over any fixed finite field (assuming Rota's conjecture holds). We prove that it is not possible to finitely axiomatize representability, or representability over any fixed infinite field,…
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…
We prove there is no sentence in the monadic second-order language MS0 that characterises when a matroid is representable over at least one field, and no sentence that characterises when a matroid is K-representable, for any infinite field K. By way of contrast, because Rota's Conjecture is true, there is a sentence that characterises F-representable matroids, for any finite field F.
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。
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