In this note, we study a constrained independent set problem for matroids and certain generalizations. The basic problem can be regarded as an ordered version of the matroid parity problem. By a reduction of this problem to matroid intersection, we prove a min-max formula. Studying the weighted case and a delta-matroid generalization, we prove that some of them are not more complex than matroid...

We discuss the relationship between matroid rank functions and a concept of discrete concavity called M-concavity. It is known that a matroid rank function and its weighted version called a weighted rank function are M-concave functions, while the (weighted) sum of matroid rank functions is not M-concave in general. We present a sufficient condition for a weighted sum of matroid rank functions ...

Frame matroids and lifted-graphic matroids are two interesting generalizations of graphic matroids. Here we introduce a new generalization, 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 mat...

For each non-negative integer k, we provide all outerplanar obstructions for the class of graphs whose cycle matroid has pathwidth at most k. Our proof combines a decomposition lemma for proving lower bounds on matroid pathwidth and a relation between matroid pathwidth and linearwidth. Our results imply the existence of a linear algorithm that, given an outerplanar graph, outputs its matroid pa...

3 In this paper, we give a complete characterization of binary matroids 4 with no P9-minor. A 3-connected binary matroid M has no P9-minor 5 if and only if M is one of the internally 4-connected non-regular minors 6 of a special 16-element matroid Y16, a 3-connected regular matroid, a 7 binary spike with rank at least four, or a matroid obtained by 3-summing 8 copies of the Fano matroid to a 3-...

L. Lovász has shown in [9] that Sperner’s combinatorial lemma admits a generalization involving a matroid defined on the set of vertices of the associated triangulation. We prove that Ky Fan’s theorem admits an oriented matroid generalization of similar nature (Theorem 3.1). Classical Ky Fan’s theorem is obtained as a corollary if the underlying oriented matroid is chosen to be the alternating ...

It is known that, in general, the coboundary polynomial and the Möbius polynomial of a matroid do not determine each other. Less is known about more specific cases. In this paper, we will investigate if it is possible that the Möbius polynomial of a matroid, together with the Möbius polynomial of the dual matroid, define the coboundary polynomial of the matroid. In some cases, the answer is aff...

In [7, 10], the concept of pseudomatroid was developed as a proper generalization of the concept of matroid. The same concept was independently developed as ∆-matroid in [4, 5]. Throughout the paper, we use the more popular name ∆-matroid for this structure. In [6], the concept of ∆-matroid was further generalized to jump system. Further interesting results on jump system are reported in [1, 3,...

By combining the concepts of graviton and matroid, we outline a new gravitational theory which we call gravitoid theory. The idea of this theory emerged as an attempt to link the mathematical structure of matroid theory with M-theory. Our observations are essentially based on the formulation of matroid bundle due to MacPherson and Anderson-Davis. Also, by considering the oriented matroid theory...