Let M be a class of matroids representable over a field F. A matroid N # M stabilizes M if, for any 3-connected matroid M # M, an F-representation of M is uniquely determined by a representation of any one of its N-minors. One of the main theorems of this paper proves that if M is minor-closed and closed under duals, and N is 3-connected, then to show that N is a stabilizer it suffices to check...

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...

A nongraphic matroid M is said to be almost-graphic if, for all elements e, either M\e or M/e is graphic. We determine completely the class of almost-graphic matroids, thereby answering a question posed by Oxley in his book “Matroid Theory.” A nonregular matroid is said to be almost-regular if, for all elements e, either M\e or M/e is regular. An element e for which both M\e and M/e are regular...

We study the lattice lat(M) of cocycles of a binary matroid M. By an isomorphism we show that such lattices are equivalent to lattices generated by the columns of proper submatrices of Sylvester matrices of full row length. As an application we show that the cocycle lattice of a recursively deened class of matroids, including all binary matroids of rank four, always has a basis consisting of co...

For any positive integer n, let [n] denote the set {1, . . . , n}, and letMn be the set of all matroids on [n]. Throughout this paper all matroids will have ground set [n], and we shall frequently omit the symbol n from our notation. Define a partial ordering on Mn by M ′ ≤ M if M ′ is a quotient of M . Let Ωn be the simplicial complex of chains in Mn ; every simplex s ∈ Ωn can be written as s ...

We continue the study initiated in [LS] of the lattice (grid) generated by the incidence vectors of cocycles of a binary matroid and its dual lattice. In [LS], we proved that every denominator in the dual lattice is a power of 2, and characterized those binary matroids M for which the largest exponent k(M) is 1. In this paper, we characterize the matroids with k(M) = 2 and, for each constant k,...

Finally, Theorem 1.2 is quite straightforward to deduce from Corollary 5.4. Alternatively , it is not diicult to prove this theorem directly using Theorem 2.1. We conclude by presenting the rst of these proofs. Proof of Theorem 1.2. From the list given in Corollary 5.4 of binary 3{connected matroids whose set of non-essential elements has rank two, we eliminate the ma-troids under (i) and (iii)...

A jump system is defined as a set of integer points (vectors) with certain exchange property, generalizing the concepts matroids, delta-matroids, and base polyhedra integral polymatroids (or submodular systems). discrete convexity concept for functions on constant-parity systems it has been used in graph theory algebra. In this paper we call “jump M-convexity” extend to M♮-convexity” larger cla...

By fuzzifying the number of occurrences of an element of a multiset, we obtain a new fuzzy structure; similarly, by fuzzifying the number of occurrences of an element of a hybrid set, we obtain another new fuzzy structure. The aim of the present work is twofold: to provide a concise definition of these new fuzzy structures and their properties and to apply them in natural computing. More specif...

D. Molodtsov (1999) introduced the concept of a soft set as a new approach for modeling uncertainties. The aim of this work is to define special kinds of soft sets, namely soft, L-fuzzifying soft, L-soft, and L-fuzzy soft neighborhood sets and to use them in order to give an alternative characterization of categories related to topology: crisp topological, L-topological, L-fuzzifying topologica...