If S is a set of matroids, then the matroid M is S-fragile if, for every element e ∈ E(M), either M\e or M/e has no minor isomorphic to a member of S. Excluded-minor characterizations often depend, implicitly or explicitly, on understanding classes of fragile matroids. In certain cases, when M is a minor-closed class of S-fragile matroids, and N ∈M, the only members ofM that contain N as a mino...

We show for each positive integer a that, if M is a minor-closed class of matroids not containing all rank-(a+ 1) uniform matroids, then there exists an integer c such that either every rank-r matroid in M can be covered by at most r rank-a sets, or M contains the GF(q)-representable matroids for some prime power q and every rank-r matroid inM can be covered by at most cq rank-a sets. In the la...

This paper considers fuzzifying topologies, a special case of I-fuzzy topologies (bifuzzy topologies), introduced by Ying [25]. It investigates topological notions defined by means of  -open sets when these are planted into the framework of Ying’s fuzzifying topological spaces (by Łukasiewicz logic in [0, 1]). In this paper we introduce some sorts of operations, called general fuzzifying opera...

Independent sets play an important role in matroid theory. In this paper, the definitions of pre-independent fuzzy set system and independent fuzzy set system in L-fuzzy setting are presented. Independent M-fuzzifying set system is introduced and some of its properties are discussed. Further independent (L,M)-fuzzy set system is given and some of its properties are obtained. The relations of th...

We study the family of matroids that do not contain the non{Fano or its dual as a minor. The main results concern the behavior of U 2;4 {minors of matroids in this family. In particular, we prove that, for any connected matroid M, there are just nitely many minor{minimal matroids in the family that contain both an M{minor and a U 2;4 {minor.

For a minor-closed class M of matroids, we let h(k) denote the maximum number of elements of a simple rank-k matroid in M. We prove that, if M does not contain all simple rank-2 matroids, then h(k) grows either linearly, quadratically, or exponentially.

In this paper, the theory of γ -convergence and γ c -convergence on nets and filters is established in fuzzifying topology. Some important and interesting results in fuzzifying topology are obtained by means of the theory.

Given matroids M and N there are two operations M ⊕2 N and M ⊗ N. When M and N are the cycle matroids of planar graphs these operations have interesting interpretations on the corresponding link diagrams. In fact, given a planar graph there are two well-established methods of generating an alternating link diagram, and in each case the Tutte polynomial of the graph is related to polynomial inva...

The cycles of a graph give natural cyclic ordering to their edge-sets, and these orderings are consistent in that two edges adjacent one cycle if only they every which appear together. An orderable matroid is whose set circuits admits such ordering. In this paper, we consider the question determining matroids orderable. Although able answer for non-binary matroids, it remains open binary matroi...

The concepts of fuzzy e-open sets and fuzzy e-continuous are introduced and studied in fuzzifying topology and by making use of these concepts, we introduce and study T e 0 -, R e 0-, T e 1 -, R e 1-, T e 2 -, T e 3 -, T e 4 -, strong T e 3 and strong T e 4 separation axioms in fuzzifying topology and give some of their characterizations as well as the relations of these axioms and other separa...