in this paper the concepts of fuzzifying β − irresolute functions and fuzzifying β − compactspaces are characterized in terms of fuzzifying β − open sets and some of their properties are discussed.

Let M be a matroid on E ∪ {`}, where ` 6∈ E is a distinguished element of M . The `-port of M is the set P = {P : P ⊆ E with P ∪ {`} a circuit of M}. Let A be the P-E incidence matrix. Let U2,4 be the uniform matroid on four elements of rank two, F7 be the Fano matroid, F ∗ 7 be the dual of F7, and F 7 be the unique series extension of F7. In this paper, we prove that the system Ax ≥ 1, x ≥ 0 i...

Bixby and Cunningham showed that a 3-connected binary matroid M is graphic if and only if every element belongs to at most two non-separating cocircuits. Likewise, Lemos showed that such a matroid M is graphic if and only if it has exactly r(M) + 1 nonseparating cocircuits. Hence the presence inM of either an element in at least three non-separating cocircuits, or of at least r(M) + 2 non-separ...

Let M be a finite matroid with rank function r. We will write A M when we mean that A is a subset of the ground set of M, and write M|A and M A for the matroids obtained by restricting M to A and contracting M on A respectively. Let M* denote the dual matroid to M. (See [1] for definitions). The main theorem is Theorem 1. The Tutte polynomial TM(x, y) satisfies TM(x, y)= : A M TM|A(0, y) TM A(x...

In this paper, we prove that an element splitting operation by every pair of elements on a cographic matroid yields a cographic matroid if and only if it has no minor isomorphic to M(K4).

In this note we investigate some matroid minor structure results. In particular, we present sufficient conditions, in terms of triangles, for a matroid to have either U2,4 or F7 or M(K5) as a minor.

Jensen and Toft [10] conjectured that every 2-edge-connected graph without a K5minor has a nowhere zero 4-flow. Walton and Welsh [24] proved that if a coloopless regular matroid M does not have a minor in {M(K3,3),M(K5)}, then M admits a nowhere zero 4-flow. In this note, we prove that if a coloopless regular matroid M does not have a minor in {M(K5),M(K5)}, then M admits a nowhere zero 4-flow....

We prove that for every proper minor-closed class M of Fp-representable matroids, there exists a O(1)-competitive algorithm for the matroid secretary problem on M. This result relies on the extremely powerful matroid minor structure theory being developed by Geelen, Gerards and Whittle. We also note that for asymptotically almost all matroids, the matroid secretary algorithm that selects a rand...

Fix two lattice paths P and Q from (0, 0) to (m, r) that use East and North steps with P never going above Q. Bonin et al. in [1] show that the lattice paths that go from (0, 0) to (m, r) and remain bounded by P and Q can be identified with the bases of a particular type of transversal matroid, which we call it a lattice path matroid. In this paper, we consider properties of lattice path matroi...

An elementary lift of a binary matroid M that arises from coextension can easily be obtained by applying the splitting operation on M. This graphic may not produce matroid. We give method to determine forbidden minors for class matroids such any set k elements is again Using this method, we obtain k=2,3,4. One compute k≥5. As consequence, whose all lifts via coextensions are also graphic. There...