Consider a binary matroid M given by its matrix representation. We show that if M is a lift of a graphic or a cographic matroid, then in polynomial time we can either solve the single commodity flow problem for M or find an obstruction for which the Max-Flow MinCut relation does not hold. The key tool is an algorithmic version of Lehman’s Theorem for the set covering polyhedron.

Let M to be a matroid defined on a finite set E and L ⊂ E. L is locked in M if M |L and M∗|(E\L) are 2-connected, and min{r(L), r∗(E\L)} ≥ 2. In this paper, we prove that the nontrivial facets of the bases polytope of M are described by the locked subsets. We deduce that finding the maximum–weight basis ofM is a polynomial time problem for matroids with a polynomial number of locked subsets. Th...

We verify a conjecture of P. Seymour (Europ. J. Combinatorics 2, p. 289) regarding circuits of a binary matroid. A circuit cover of a integer-weighted matroid (M; p) is a list of circuits of M such that each element e is in exactly p(e) circuits from the list. We characterize those binary matroids for which two obvious necessary conditions for a weighting (M; p) to have a circuit cover are also...

A 3-connected matroid M is sequential or has path width 3 if its ground set E(M) has a sequential ordering, that is, an ordering (e1, e2, . . . , en) such that ({e1, e2, . . . , ek}, {ek+1, ek+2, . . . , en}) is a 3-separation for all k in {3, 4, . . . , n − 3}. In this paper, we consider the possible sequential orderings that such a matroid can have. In particular, we prove that M essentially ...

Let M be a matroid on [n] and E be the graded algebra generated over a field k generated by the elements 1, e1, . . . , en . Let =(M) be the ideal of E generated by the squares e2 1, . . . , e 2 n , elements of the form ei e j + ai j e j ei and ‘boundaries of circuits’, i.e., elements of the form ∑ χ j ei1 . . . ei j−1 ei j+1 . . . eim , with χ j ∈ k and ei1 , . . . , eim a circuit of the matro...

In this paper we study the question of existence of a cycle basis (that is, a basis consisting only of cycles) for the lattice Z(M) generated by the cycles of a binary matroid M. We show that, if M has no Fano dual minor, then the set of fundamental circuits of any maximal independent set of M can be completed to a cycle basis of Z(M); moreover, for any one-element extension M 0 of such matroid...

In this paper we construct from a cographic matroid M , a pure multicomplex whose degree sequence is the h–vector of the the matroid complex of M. This result proves a conjecture of Richard Stanley [Sta96] in the particular case of cographic matroids. We also prove that the multicomplexes constructed are M–shellable, so proving a conjecture of Manoj Chari [Cha97] again in the case of cographic ...

Generalizing polynomials previously studied in the context of linear codes, we define weight polynomials and an enumerator for a matroid M . Our main result is that these polynomials are determined by Betti numbers associated with N0-graded minimal free resolutions of the Stanley-Reisner ideals of M and so-called elongations of M . Generalizing Greene’s theorem from coding theory, we show that ...

We study the Bergman complex B(M) of a matroid M : a polyhedral complex which arises in algebraic geometry, but which we describe purely combinatorially. We prove that a natural subdivision of the Bergman complex of M is a geometric realization of the order complex of its lattice of flats. In addition, we show that the Bergman fan B̃(Kn) of the graphical matroid of the complete graph Kn is homeo...

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...