We study characterisations of strong $Δ$-matroids, compiling a list of five equivalent descriptions. We show a variant of Wenzel's exchange property and the hyperplane exchange property of Borovik-Gelfand-White are equivalent. We also introduce two novel characterisations in terms of 'peerless' and 'isolated' antipodes within the system of feasible sets, banning certain configurations of antipodes either globally or locally. As a corollary, we obtain new 'local' exchange axioms for matroids and $Δ$-matroids. We give algebraic motivation for these new characterisations by introducing the peerless antipode equations, tropical equations that govern whether a $Δ$-matroid has no peerless antipodes. We show that these arise as the tropicalisation of a specific basis of quadratics cutting out the orthogonal Grassmannian.
A simple graph $G=(V,E)$ is a $(2,1)$-circuit if $|E|=2|V|$ and $|E(H)|\leq 2|V(H)|-1$ for every proper subgraph $H$ of $G$. Motivated, in part, by ongoing work to understand unique realisations of graphs on surfaces, we derive a constructive characterisation of $(2,1)$-circuits. The characterisation uses the well known 1-extension and $X$-replacement operations as well as several summation moves to glue together $(2,1)$-circuits over small cutsets.
We conjecture that the class of frame matroids can be characterised by a sentence in the monadic second-order logic of matroids, and we prove that there is such a characterisation for the class of bicircular matroids. The proof does not depend on an excluded-minor characterisation.
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 $X \subsetneq V$ 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 join operations to complete the characterisation. Extensions are discussed to when the sparsity matroid is connected and this is applied to the theory of frameworks on surfaces to provide a conjectured characterisation of when fram…
The Las Vergnas' strong map conjecture, states that any strong map of oriented matroids $f:\mathcal{M}_1\rightarrow\mathcal{M}_2$ can be factored into extensions and contractions. The conjecture is known to be false due to a construction by Richter-Gebert, he find a non-factorizable strong map $f:\mathcal{M}_1\rightarrow\mathcal{M}_2$, however in his example $\mathcal{M}_1$ is not realizable. The problem that whether there exists a non-factorizable strong map between realizable oriented matroids still remains open. In this paper we provide a counterexample to the strong map conjecture on reali…
We investigate the strong Rayleigh property of matroids for which the basis enumerating polynomial is invariant under a Young subgroup of the symmetric group on the ground set. In general, the Grace-Walsh-Szegő theorem can be used to simplify the problem. When the Young subgroup has only two orbits, such a matroid is strongly Rayleigh if and only if an associated univariate polynomial has only real roots. When this polynomial is quadratic we get an explicit structural criterion for the strong Rayleigh property. Finally, if one of the orbits has rank two then the matroid is strongly Rayleigh if…
We use the Strong Splitter Theorem to decompose the excluded minor class of binary matroids with no $E_4$-minor. Using this theorem we can get the 3-decomposers and the extremal internally 4-connected matroids as well as any other important matroids in the class. The matroid $E_4$ is a self-dual 10-element binary 3-connected matroid that plays a useful role in structural results. It is a single-element coextension of $P_9$, which is a single-element extension of the 4-wheel. We show that the extremal matroids in this class are the binary rank-$r$ spikes $Z_r$, the rank 3 and 4 projective geome…
The Splitter Theorem states that, if $N$ is a 3-connected proper minor of a 3-connected matroid $M$ such that, if $N$ is a wheel or whirl then $M$ has no larger wheel or whirl, respectively, then there is a sequence $M_0,..., M_n$ of 3-connected matroids with $M_0\cong N$, $M_n=M$ and for $i\in \{1,..., n\}$, $M_i$ is a single-element extension or coextension of $M_{i-1}$. Observe that there is no condition on how many extensions may occur before a coextension must occur. In this paper, we give a strengthening of the Splitter Theorem, as a result of which we can obtain, up to isomorphism, $M$ …
The multiple exchange property for matroid bases is generalized for valuated matroids and M$^\natural$-concave set functions. The proof is based on the Fenchel-type duality theorem in discrete convex analysis. The present result has an implication in economics: The strong no complementarities (SNC) condition of Gul and Stacchetti is in fact equivalent to the gross substitutes (GS) condition of Kelso and Crawford.
This paper investigates infinite matroids from a categorical perspective. We prove that the category of infinite matroids is a proto-exact category in the sense of Dyckerhoff and Kapranov, thereby generalizing our previous result on the category of finite matroids. We also characterize finitary matroids as co-limits of finite matroids, and show that the finite matroids are precisely the finitely presentable objects in this category.
The multiple exchange property for matroid bases has recently been generalized for valuated matroids and M$^{\natural}$-concave set functions. This paper establishes a stronger form of this multiple exchange property that imposes a cardinality condition on the exchangeable subset. The stronger form immediately implies the defining exchange property of M$^{\natural}$-concave set functions, which was not the case with the recently established multiple exchange property without the cardinality condition.
We present an algebraic framework which simultaneously generalizes the notion of linear subspaces, matroids, valuated matroids, oriented matroids, and regular matroids. To do this, we first introduce algebraic objects called tracts which generalize both hyperfields in the sense of Krasner and partial fields in the sense of Semple and Whittle. We then define matroids over tracts; in fact, there are (at least) two natural notions of matroid in this general context, which we call weak and strong matroids. We give "cryptomorphic" axiom systems for such matroids in terms of circuits, Grassmann-Plüc…
An integer additive set-indexer is defined as an injective function $f:V(G)\rightarrow 2^{\mathbb{N}_0}$ such that the induced function $g_f:E(G) \rightarrow 2^{\mathbb{N}_0}$ defined by $g_f (uv) = f(u)+ f(v)$ is also injective, where $f(u)+f(v)$ is the sumset of $f(u)$ and $f(v)$. If $g_f(uv)=k~\forall~uv\in E(G)$, then $f$ is said to be a $k$-uniform integer additive set-indexers. An integer additive set-indexer $f$ is said to be a strong integer additive set-indexer if $|g_f(uv)|=|f(u)|.|f(v)|~\forall ~ uv\in E(G)$. We already have some characteristics of the graphs which admit strong inte…
We present an algebraic framework which simultaneously generalizes the notion of linear subspaces, matroids, valuated matroids, and oriented matroids. We call the resulting objects matroids over hyperfields. In fact, there are (at least) two natural notions of matroid in this context, which we call weak and strong matroids. We give "cryptomorphic" axiom systems for such matroids in terms of circuits, Grassmann-Plucker functions, and dual pairs, and establish some basic duality theorems. We also show that if F is a doubly distributive hyperfield then the notions of weak and strong matroid over …
The structure of the category of matroids and strong maps is investigated: it has coproducts and equalizers, but not products or coequalizers; there are functors from the categories of graphs and vector spaces, the latter being faithful; there is a functor to the category of geometric lattices, that is nearly full; there are various adjunctions and free constructions on subcategories, inducing a simplification monad; there are two orthogonal factorization systems; some, but not many, combinatorial constructions from matroid theory are functorial.
The basis exchange axiom has been a driving force in the development of matroid theory. However, the axiom gives only a local characterization of the relation of bases, which is a major stumbling block to further progress, and providing a global understanding of the structure of matroid bases is a fundamental goal in matroid optimization. While studying the structure of symmetric exchanges, Gabow proposed the problem that any pair of bases admits a sequence of symmetric exchanges. A different extension of the exchange axiom was proposed by White, who investigated the equivalence of compatible …
Two pairs of disjoint bases $\mathbf{P}_1=(R_1,B_1)$ and $\mathbf{P}_2=(R_2,B_2)$ of a matroid $M$ are called equivalent if $\mathbf{P}_1$ can be transformed into $\mathbf{P}_2$ by a series of symmetric exchanges. In 1980, White conjectured that such a sequence always exists whenever $R_1\cup B_1=R_2\cup B_2$. A strengthening of the conjecture was proposed by Hamidoune, stating that minimum length of an exchange is at most the rank of the matroid. We propose a weighted variant of Hamidoune's conjecture, where the weight of an exchange depends on the weights of the exchanged elements. We prove …
Anari, Gharan, and Vinzant proved (complete) log-concavity of the basis generating functions for all matroids. From the viewpoint of combinatorial Hodge theory, it is natural to ask whether these functions are "strictly" log-concave for simple matroids. In this paper, we show this strictness for simple graphic matroids, that is, we show that Kirchhoff polynomials of simple graphs are strictly log-concave. Our key observation is that the Kirchhoff polynomial {of a complete graph} can be seen as the (irreducible) relative invariant of a certain prehomogeneous vector space, which may be independe…
A polynomial f is said to have the half-plane property if there is an open half-plane H, whose boundary contains the origin, such that f is non-zero whenever all the variables are in H. This paper answers several open questions regarding multivariate polynomials with the half-plane property and matroid theory. * We prove that the support of a multivariate polynomial with the half-plane property is a jump system. This answers an open question posed by Choe, Oxley, Sokal and Wagner and generalizes their recent result claiming that the same is true whenever the polynomial is also homogeneous. * W…
