A Linear Programming Construction of Fully Optimal Bases in Graphs and Hyperplane Arrangements

The fully optimal basis of a bounded acyclic oriented matroid on a linearly ordered set has been defined and studied by the present authors in a series of papers, dealing with graphs, hyperplane arrangements, and oriented matroids (in order of increasing generality). This notion provides a bijection between bipolar orientations and uniactive internal spanning trees in a graph resp. bounded regions and uniactive internal bases in a hyperplane arrangement or an oriented matroid (in the sense of Tutte activities). This bijection is the basic case of a general activity preserving bijection between reorientations and subsets of an oriented matroid, called the active bijection, providing bijective versions of various classical enumerative results.Fully optimal bases can be considered as a strenghtening of optimal bases from linear programming, with a simple combinatorial definition. Our first construction used this purely combinatorial characterization, providing directly an algorithm to compute in fact the reverse bijection. A new definition uses a direct construction in terms of a linear programming. The fully optimal basis optimizes a sequence of nested faces with respect to a sequence of objective functions (whereas an optimal basis in the usual sense optimizes one vertex with respect to one objective function). This note presents this construction in terms of graphs and linear algebra.     

Fully Optimal Bases and the Active Bijection in Graphs,Hyperplane Arrangements,and Oriented Matroids

In this note, we present the main results of a series of forthcoming papers, dealing with bi-jective generalizations of some counting formulas. New intrinsic constructions in oriented matroids on a linearly ordered set of elements establish notably structural links between counting regions and linear programming. We introduce fully optimal bases, which have a simple combinatorial characterization, and strengthen the well-known optimal bases of linear programming. Our main result is that every bounded region of an ordered hyperplane arrangement, or ordered oriented matroid, has a unique fully optimal basis, providing the active bijection between bounded regions and uniactive internal bases. The active bijec-tion is extended to an activity preserving mapping between all reorientations and all bases of an ordered oriented matroid. It gives a bijective interpretation of the equality of two expressions for the Tutte polynomial, as well as a new expression of this polynomial in terms of beta invariants of minors. There are several refinements, such as an activity preserving bijection between regions (acyclic reorientations) and no-broken-circuit subsets, and others in terms of hyperplane arrangements, graphs, and permutations.     

On Tutte polynomial expansion formulas in perspectives of matroids and oriented matroids

We introduce the active partition of the ground set of an oriented matroid perspective (or morphism, or quotient, or strong map) on a linearly ordered ground set. The reorientations obtained by arbitrarily reorienting parts of the active partition share the same active partition. This yields an equivalence relation for the set of reorientations of an oriented matroid perspective, whose classes are enumerated by coefficients of the Tutte polynomial, and a remarkable partition of the set of reorientations into Boolean lattices, from which we get a short direct proof of a 4-variable expansion formula for the Tutte polynomial in terms of orientation activities. This formula was given in the last unpublished preprint by Michel Las Vergnas; the above equivalence relation and notion of active partition generalize a former construction in oriented matroids by Michel Las Vergnas and the author; and the possibility of such a proof technique in perspectives was announced in the aforementioned preprint. We also briefly highlight how the 5-variable expansion of the Tutte polynomial in terms of subset activities in matroid perspectives comes in a similar way from the known partition of the power set of the ground set into Boolean lattices related to subset activities (and we complete the proof with a property which was missing in the literature). In particular, the paper applies to matroids and oriented matroids on a linearly ordered ground set, and applies to graph and directed graph homomorphisms on a linearly ordered edge-set.     

Bijections between affine hyperplane arrangements and valued graphs

We show new bijective proofs of previously known formulas for the number of regions of some deformations of the braid arrangement, by means of a bijection between the no-broken-circuit sets of the corresponding integral gain graphs and some kinds of labelled binary trees. This leads to new bijective proofs for the Shi, Catalan, and similar hyperplane arrangements.     

On the Bounded Complex of an Affine Oriented Matroid

We prove that the bounded complex of an affine oriented matroid is pure and collapsible. We also generalize Zaslavsky's central decomposition theorem for hyperplane arrangements to affine oriented matroids.     

Extension spaces of oriented matroids

We study the space of all extensions of a real hyperplane arrangement by a new pseudohyperplane, and, more generally, of an oriented matroid by a new element. The question whether this space has the homotopy type of a sphere is a special case of the “Generalized Baues Problem” of Billera, Kapranov, and Sturmfels, via the Bohne-Dress theorem on zonotopal tilings. We prove that the extension space is spherical for the class of strongly euclidean oriented matroids. This class includes the alternating matroids and all oriented matroids of rank at most 3 or of corank at most 2. In general it is not known whether the extension space is connected for all realizable oriented matroids (hyperplane arrangements). We show that the subspace of realizable extensions is always connected but not necessarily spherical. Nonrealizable oriented matroids of rank 4 with disconnected extension spaces were recently constructed by Mnëv and Richter-Gebert.     

Tutte-martin polynomials and orienting vectors of isotropic systems

Isotropic systems are structures which unify some properties of 4-regular graphs and of pairs of dual binary matroids. In this paper we unify the properties of the symmetric Tutte polynomials (i.e. with equal variables) of binary matroids and of the Martin polynomials of 4-regular graphs. For this purpose we introduce the orienting vectors of an isotropic system in order to generalize the eulerian orientations of 4-regular graphs.     

The sand-pile model and Tutte polynomials

A new explicit bijection between spanning trees and recurrent configurations of the sand-pile model is given. This mapping is such that the difference between the number of grains on a configuration and the external activity of the associate tree is the number of edges of the graph. It gives a bijective proof of a result of Merino López expressing the generating function of recurrent configurations as an evaluation of the Tutte polynomial.     

The Las Vergnas polynomial for embedded graphs

The Las Vergnas polynomial is an extension of the Tutte polynomial to cellularly embedded graphs. It was introduced by Michel Las Vergnas in 1978 as special case of his Tutte polynomial of a morphism of matroids. While the general Tutte polynomial of a morphism of matroids has a complete set of deletion–contraction relations, its specialisation to cellularly embedded graphs does not. Here we extend the Las Vergnas polynomial to graphs in pseudo-surfaces. We show that in this setting we can define deletion and contraction for embedded graphs consistently with the deletion and contraction of the underlying matroid perspective, thus yielding a version of the Las Vergnas polynomial with complete recursive definition. This also enables us to obtain a deeper understanding of the relationships among the Las Vergnas polynomial, the Bollobás–Riordan polynomial, and the Krushkal polynomial. We also take this opportunity to extend some of Las Vergnas’ results on Eulerian circuits from graphs in surfaces of low genus to graphs in surfaces of arbitrary genus.     

The bounded complex of a uniform affine oriented matroid is a ball

Zaslavsky conjectures that the bounded complex of a simple hyperplane arrangement is homeomorphic to a ball. We prove this conjecture for the more general uniform affine oriented matroids.     

What is a complex matroid?

Following an “ansatz” of Björner and Ziegler [BZ], we give an axiomatic development of finite sign vector systems that we callcomplex matroids. This includes, as special cases, the sign vector systems that encode complex arrangements according to [BZ], and the complexified oriented matroids, whose complements were considered by Gel'fand and Rybnikov [GeR]. Our framework makes it possible to study complex hyperplane arrangements as entirely combinatorial objects. By comparing complex matroids with 2-matroids, which model the more general 2-arrangements introduced by Goresky and MacPherson [GoM], the essential combinatorial meaning of a “complex structure” can be isolated. Our development features a topological representation theorem for 2-matroids and complex matroids, and the computation of the cohomology of the complement of a 2-arrangement, including its multiplicative structure in the complex case. Duality is established in the cases of complexified oriented matroids, and for realizable complex matroids. Complexified oriented matroids are shown to be matroids with coefficients in the sense of Dress and Wenzel [D1], [DW1], but this fails in general.     

Convexity in oriented matroids

We generalize to oriented matroids classical notions of Convexity Theory: faces of convex polytopes, convex hull, etc., and prove some basic properties. We relate the number of acyclic orientations of an orientable matroid to an evaluation of its Tutte polynomial.     

Interlacement and activities in delta-matroids

We generalize theories of graph, matroid, and ribbon-graph activities to delta-matroids. As a result, we obtain an activity based feasible-set expansion for a transition polynomial of delta-matroids defined by Brijder and Hoogeboom. This result yields feasible-set expansions for the two-variable Bollobás–Riordan and interlace polynomials of a delta-matroid. In the former case, the expansion obtained directly generalizes the activities expansions of the Tutte polynomial of graphs and matroids.     

Invariants of Composite Networks Arising as a Tensor Product

We generalize Brylawski’s formula of the Tutte polynomial of a tensor product of matroids to colored connected graphs, matroids, and disconnected graphs. Unlike the non-colored tensor product where all edges have to be replaced by the same graph, our colored generalization of the tensor product operation allows individual edge replacement. The colored Tutte polynomials we compute exists by the results of Bollobás and Riordan. The proof depends on finding the correct generalization of the two components of the pointed Tutte polynomial, first studied by Brylawski and Oxley, and on careful enumeration of the connected components in a tensor product. Our results make the calculation of certain invariants of many composite networks easier, provided that the invariants are obtained from the colored Tutte polynomials via substitution and the composite networks are represented as tensor products of colored graphs. In particular, our method can be used to calculate (with relative ease) the expected number of connected components after an accident hits a composite network in which some major links are identical subnetworks in themselves.      

Orientations, Lattice Polytopes, and Group Arrangements I: Chromatic and Tension Polynomials of Graphs

This is the first one of a series of papers on association of orientations, lattice polytopes, and group arrangements to graphs. The purpose is to interpret the integral and modular tension polynomials of graphs at zero and negative integers. The whole exposition is put under the framework of subgroup arrangements and the application of Ehrhart polynomials. Such a viewpoint leads to the following main results of the paper: (i) the reciprocity law for integral tension polynomials; (ii) the reciprocity law for modular tension polynomials; and (iii) a new interpretation for the value of the Tutte polynomial T(G; x, y) of a graph G at (1, 0) as the number of cut-equivalence classes of acyclic orientations on G.     

The Topological Representation of Oriented Matroids

We present a new proof of the Topological Representation Theorem for oriented matroids in the general rank case. Our proof is based on an earlier rank 3 version. It uses hyperline sequences and the generalized Schonflies theorem. As an application, we show that one can read off oriented matroids from arrangements of embedded spheres of codimension one, even if wild spheres are involved.     

Connectivity in bicircular matroids

Tutte has defined n-connection for matroids and proved a connected graph is n-connected if and only if its polygon matroid is n-connected. In this paper we introduce a new notion of connection in graphs, called n-biconnection, and prove an analogous theorem for graphs and their bicircular matroids. Results concerning 3-biconnected graphs are also presented.     

On the colored Tutte polynomial of a graph of bounded treewidth

We observe that a formula given by Negami [Polynomial invariants of graphs, Trans. Amer. Math. Soc. 299 (1987) 601-622] for the Tutte polynomial of a k-sum of two graphs generalizes to a colored Tutte polynomial. Consequently, an algorithm of Andrzejak [An algorithm for the Tutte polynomials of graphs of bounded treewidth, Discrete Math. 190 (1998) 39-54] may be directly adapted to compute the colored Tutte polynomial of a graph of bounded treewidth in polynomial time. This result has also been proven by Makowsky [Colored Tutte polynomials and Kauffman brackets for graphs of bounded tree width, Discrete Appl. Math. 145 (2005) 276-290], using a different algorithm based on logical techniques.     

A Bijection for Eulerian-equivalence Classes of Totally Cyclic Orientations

Gioan showed that the number of cycle reversing classes of totally cyclic orientations of a given graph can be calculated as an evaluation of the corresponding Tutte polynomial. We note that the concept of cycle reversing classes of orientations coincides with that of Eulerian-equivalence classes considered by Chen and Stanley, and Kochol. Based on this coincidence, we give a bijective proof of Gioan’s result. Precisely, the main result of the paper is an algorithmic bijection between the set of Eulerian-equivalence classes of totally cyclic orientations and the set of spanning trees without internally active edges.