We express the matroid polytope PM of a matroid M as a signed Minkowski sum of simplices, and obtain a formula for the volume of PM . This gives a combinatorial expression for the degree of an arbitrary torus orbit closure in the Grassmannian Grk,n. We then derive analogous results for the independent set polytope and the associated flag matroid polytope of M . Our proofs are based on a natural...

Rough set is a theory of data analysis and a mathematical tool for dealing with vagueness, incompleteness, and granularity. Matroid, as a branch of mathematics, is a structure that generalizes linear independence in vector spaces. In this paper, we establish the relationships between partition matroids and rough sets through k-rank matroids. On the one hand, k-rank matroids are proposed to repr...

The isotropic matroid M [IAS(G)] of a graph G is a binary matroid, which is equivalent to the isotropic system introduced by Bouchet. In this paper we discuss four notions of connectivity related to isotropic matroids and isotropic systems. We show that the isotropic system connectivity defined by Bouchet is equivalent to vertical connectivity of M [IAS(G)], and if G has at least four vertices,...

This paper contains a survey of the A-theory of simplicial complexes and graphs, a combinatorial homotopy theory developed recently.The initialmotivation arises from the use of simplicial complexes as models for a variety of complex systems and their dynamics. This theory diverges from classical homotopy theory in several crucial aspects. It is related to prior work in matroid theory, graph the...

Define a subset S ~ P(X) to be independent if and only if {z~[BC S} is a linearly independent subset of V. This clearly induces a matroid structure on P(X). This matroid has been studied in connection with many problems in the theory of designs or extremal problems for families of finite sets. In fact many interesting inequalities are proved by showing that certain families of subsets of X corr...

The well-studied Vámos matroid has provided a wealth of interesting theoretical results in matroid theory. We use the Vámos matroid to construct a new network, which we call the Vámos network. We then exploit the Vámos network to answer in the negative the open question as to whether Shannon-type information inequalities are in general sufficient for computing network coding capacities. To acco...

This paper provides a partial generalization to matroid theory of the result of Halin and Jung that each simple graph with minimum vertex degree at least 4 has K5 or the octahedron K2,2,2 as a minor.

Mader’s disjoint S-paths problem unifies two generalizations of bipartite matching: (a) nonbipartite matching and (b) disjoint s–t paths. Lovász (1980, 1981) first proposed an efficient algorithm for this problem via a reduction to matroid matching, which also unifies two generalizations of bipartite matching: (a) non-bipartite matching and (c) matroid intersection. While the weighted versions ...

Last lecture we covered matroid intersection, and defined matroid union. In this lecture we review the definitions of matroid intersection, and then show that the matroid intersection polytope is TDI. This is Chapter 41 in Schrijver’s book. Next we review matroid union, and show that unlike matroid intersection, the union of two matroids is again a matroid. This material is largely contained in...