An element e of a 3–connected matroid M is essential if neither the deletion nor the contraction of e from M is 3–connected. Tutte’s 1966 Wheels and Whirls Theorem proves that the only 3–connected matroids in which every element is essential are the wheels and whirls. It was proved by Oxley and Wu that if a 3–connected matroid M has a non-essential element, then it has at least two such element...

Two elements in an oriented matroid are inseparable if they have either the same sign in every signed circuit containing them both or opposite signs in every signed circuit containing them both. Two elements of a matroid are adjacent if there is no M(K4)-minor using them both, and in which they correspond to a matching of K4. We prove that two elements e, { of an oriented matroid are inseparabl...

We propose a new approach for speeding up enumeration algorithms. The approach does not relies on data structures deeply, but utilizes some analysis of its computation time. The enumeration algorithms for directed spanning trees, matroid bases, and some bipartite matching problems are speeded up by this approach. For a given graph G = (V,E), the time complexity of the algorithm for directed spa...

We make a number of observations about matter-ghost string phase, which may eventually lead to a formal connection between matroid theory and string theory. In particular, in order to take advantage of the already established connection between matroid theory and Chern-Simons theory, we propose a generalization of string theory in terms of some kind of Kahler metric. We show that this generaliz...

We prove that the maximum size of a simple binary matroid of rank r ≥ 5 with no AG(3, 2)-minor is (r+1 2 ) and characterize those matroids achieving this bound. When r ≥ 6, the graphic matroid M(Kr+1) is the unique matroid meeting the bound, but there are a handful of matroids of lower ranks meeting or exceeding this bound. In addition, we determine the size function for nongraphic simple binar...

Of course, if we take S to be the set of columns of a matrix, and the independent sets to be the ones that are linearly independent, we get a first example, called a linear matroid. Another important class consists of the graphic ones – here S is the set of edges of a graph G and a subset is independent if it forms a forest. Whitney established some equivalent versions of the axioms, highlighte...

Hall’s theorem for bipartite graphs gives a necessary and sufficient condition for the existence of a matching in a given bipartite graph. Aharoni and Ziv [2] generalized the notion of matchability to a pair of possibly infinite matroids on the same set and gave a condition that is sufficient for the matchability of a given pair (M,W) of finitary matroids, where the matroid M is SCF — a sum of ...

This paper introduces a general matroid-theoretic construction which includes, as special cases, elementary lifts of matroids and bias matroids of biased graphs. To perform the construction on a matroid A/ , it is necessary (but not sufficient) to have a submodular function inducing M . Elementary lifts are obtained when the submodular function chosen is the rank function of M . We define what ...

In this paper,  it is shown that the category of $L$-ordered fuzzifying convergence spaces contains the category of pretopological $L$-ordered fuzzifying convergence spaces as a bireflective subcategory and the latter contains the category of topological $L$-ordered fuzzifying convergence spaces as a bireflective subcategory. Also, it is proved that the category of $L$-ordered fuzzifying conver...

Hugo Hadwiger proved that a graph that is not 3-colorable must have a K4minor and conjectured that a graph that is not k-colorable must have a Kk+1minor. By using the Hochstättler-Nešetřil definition for the chromatic number of an oriented matroid, we formulate a generalized version of Hadwiger’s conjecture that might hold for the class of oriented matroids. In particular, it is possible that e...