We show that, for every finite field , the class of all representable matroids of branch-width at most a constant t has a decidable MSO theory. In the other direction, we prove that every class of -representable matroids with a decidable MSO theory must have uniformly bounded branch-width.

We investigate an approach to matroid complexity that involves describing a matroid via a list of independent sets, bases, circuits, or some other family of subsets of the ground set. The computational complexity of algorithmic problems under this scheme appears to be highly dependent on the choice of input type. We define an order on the various methods of description, and we show how this ord...

Using Rado’s theorem for the existence of an independent transversal of family of subsets of a set on which a matroid is defined, we give a proof of Landau’s theorem for the existence of a tournament with a prescribed degree sequence. A similar approach is used to determine when a partial tournament can be extended to a tournament with a prescribed degree sequence. Mathematics Subject Classific...

In this paper, we study flag structures of matroid base polytopes. We describe faces of matroid base polytopes in terms of matroid data, and give conditions for hyperplane splits of matroid base polytopes. Also, we show how the cd-index of a polytope can be expressed when a polytope is split by a hyperplane, and apply these to the cd-index of a matroid base polytope of a rank 2 matroid.

Abstract We use techniques from Gromov–Witten theory to construct new invariants of matroids taking value in the Chow groups spaces rational curves permutohedral toric variety. When matroid is realizable by a complex hyperplane arrangement, our coincide with virtual fundamental classes used define logarithmic wonderful models arrangement complements, for any structure supported on boundary. bou...

We show that for any k-connected graph having cocircumference c∗, there is a cycle which intersects every cocycle of size c∗ − k+2 or greater. We use this to show that in a 2connected graph, there is a family of at most c∗ cycles for which each edge of the graph belongs to at least two cycles in the family. This settles a question raised by Oxley. A certain result known for cycles and cocycles ...

Las Vergnas’ generalizations of the Tutte polynomial are studied as follows. The theory of Tutte-Grothendieck matroid invariantsfis modified so the Tutte decomposition ,f(M) =f(M\e) +f( M/e) is applied only when e $ P (and e is neither a loop nor an isthmus) where P is a distinguished set of points called ports. The resulting “P-ported” Tutte polynomial tp has variables I, w; q,, qZ, . . . . qm...

The theory of oriented matroids provides a broad setting in which to model, describe, and analyze combinatorial properties of geometric configurations. Apparently totally different mathematical objects such as point and vector configurations, arrangements of hyperplanes, convex polytopes, directed graphs, and linear programs find a common generalization in the language of oriented matroids. The...

We consider a specialization YM (q, t) of the Tutte polynomial of a matroid M which is inspired by analogy with the Potts model from statistical mechanics. The only information lost in this specialization is the number of loops of M . We show that the coefficients of YM (1 − p, t) are very simply related to the ranks of the Whitney homology groups of the opposite partial orders of the independe...

The broken-circuit complex introduced by H. Wilf (Which polynomials are chromatic!, Proc. Colloq. Combinational Theory (Rome, 1973)) of a matroid G is shown to be a cone over a related complex, the reduced broken-circuit complex Q'(G). The topological structure of Q'(G) is studied, its Euler characteristic is computed, and joins and skeletons are shown to exist in the class of all such complexe...