A characterization of box 1/d1/d-integral binary clutters

Let Q6 denote the port of the dual Fano matroid F*7 and let Q7 denote the clutter consisting of the circuits of the Fano matroid F7 that contain a given element. Let be a binary clutter on E and let d = 2 be an integer. We prove that all the vertices of the polytope {x E+ | x(C) = 1 for C } n {x | a = x = b} are -integral, for any -integral a, b, if and only if does not have Q6 or Q7 as a minor. This includes the class of ports of regular matroids. Applications to graphs are presented, extending a result from Laurent and Pojiak [7]

The Anti-join Composition and Polyhedra

In this paper we describe a binary composition operation, the anti-join, which combines a pair of clutters L1 and L2 to give a new clutter L. The anti-join operation is, in some sense, the ''dual'' of the join operation, introduced by Cunningham [10]. In fact, it has the property that the blocker of the clutter L obtained by joining two clutters L1 and L2, is the anti-join of the blockers of L1 and L2. For such an operation we show how the linear descriptions of the polyhedra Q(L1) and Q(L2) have to be combined to produce a linear description of the polyhedron Q(L). Moreover, given a set F subset-or-equal-to {lambda: 0 less-than-or-equal-to lambda less-than-or-equal-to 1} such that {0, 1} subset-or-equal-to F and a is-an-element-o F if and only if (1-a) is-an-element-of F, we define the F-property for covering polyhedra as a proper generalization of the Fulkerson property, to which it reduces for F = {0, 1}. We prove that the anti-join operation preserves the F-property. This implies the characterization of the coefficients of the facet-defining inequalities for the cycle and cocycle polyhedra associated with graphs noncontractible to the four-wheel W4

The anti-join composition and polyhedra

AbstractGiven a clutter L, we associate with it the covering polyhedron QL, that is the dominant of the convex hull of incidence vectors of all the covers of L.In this paper we describe a binary composition operation, the anti-join, which combines a pair of clutters L1 and L2 to give a new clutter L. The anti-join operation is, in some sense, the “dual” of the join operation, introduced by Cunningham [10]. In fact, it has the property that the blocker of the clutter L obtained by joining two clutters L1 and L2, is the anti-join of the blockers of L1 and L2.For such an operation we show how the linear descriptions of the polyhedra Q(L1) and Q(L2) have to be combined to produce a linear description of the polyhedron QL.Moreover, given a set F ⊆ {λ: 0⩽λ⩽1} such that {0, 1} ⊆ F and aϵF if and only if (1−a)ϵF, we define the F-property for covering polyhedra as a proper generalization of the Fulkerson property, to which it reduces for F={0, 1}. We prove that the anti-join operation preserves the F-property. This implies the characterization of the coefficients of the facet-defining inequalities for the cycle and cocycle polyhedra associated with graphs noncontractible to the four-wheel W4