In this paper we show the rst polynomial-time algorithm for the problem of minimizing submodular functions on the product of diamonds of nite size. This submodular function minimization problem is reduced to the membership problem for an associated polyhedron, which is equivalent to the optimization problem over the polyhedron, based on the ellipsoid method. The latter optimization problem is a...

In this paper, rate 1/p binary systematic quasi-cyclic (QC) codes are constructed based on Matroid Theory (MT). The relationship between the generator matrix and minimum distance d is derived through MT, which is benefit to find numbers of QC codes with large minimum distance by our Matroid search algorithm. More than seventy of QC codes that extend previously published results are presented. A...

Let M = (E,F) be a matroid on a set E, and B one of its bases. A closed set θ ⊆ E is saturated with respect to B when |θ ∩B| = r(θ), where r(θ) is the rank of θ. The collection of subsets I of E such that |I ∩ θ| ≤ r(θ) for every closed saturated set θ turns out to be the family of independent sets of a new matroid on E, called base-matroid and denoted by MB . In this paper we prove that a grap...

We prove that, if M is a weakly 4-connected matroid with |E(M)| 7 and neither M nor M∗ is isomorphic to the cycle matroid of a ladder, then M has a proper minor M ′ such that M ′ is weakly 4-connected and |E(M ′)| |E(M)| − 2 unless M is some 12-element matroid with a special structure. © 2007 Elsevier Inc. All rights reserved.

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....

the aim of this paper is to study the categorical relations betweenmatroids, goetschel-voxman’s fuzzy matroids and shi’s fuzzifying matroids.it is shown that the category of fuzzifying matroids is isomorphic to that ofclosed fuzzy matroids and the latter is concretely coreflective in the categoryof fuzzy matroids. the category of matroids can be embedded in that offuzzifying matroids as a simul...

Let q be a power of a prime p. Matrices over Fq in which every subset of basis size of the columns are independent, are of interest in coding theory, matroid theory, and projective geometry. For any positive integer m ≤ p and bijection σ : N≤q−1 ∪ {0} → Fq, we show that the m× (q + 1) matrix Hq,m, with {Uq}i,j = 

We provide a constructive characterisation of circuits in the simple (2, 2)sparsity matroid. A circuit is a simple graph G = (V,E) with |E| = 2|V | − 1 and the number of edges induced by any X ( V is at most 2|X| − 2. Insisting on simplicity results in the Henneberg operation being enough only when the graph is sufficiently connected. Thus we introduce 3 different sum operations to complete the...

In this paper we employ Tutte’s theory of bridges to derive a decomposition theorem for binary matroids arising from signed graphs. The proposed decomposition differs from previous decomposition results on matroids that have appeared in the literature in the sense that it is not based on k-sums, but rather on the operation of deletion of a cocircuit. Specifically, it is shown that certain minor...

One of the most basic examples of matroid duality is the following. Let G be a graph imbedded in the plane and let G∗ be its topological dual graph. If M(G) is the cycle matroid of G, then the dual matroid M∗(G) = M(G∗). If G is a connected graph that is 2-cell imbedded in a surface of demigenus d > 0 (the demigenus is equal to 2 minus the euler characteristic of the surface), then M∗(G) 6= M(G...