On minor-minimally-connected matroids

AbstractBy a well-known result of Tutte, if e is an element of a connected matroid M, then either the deletion or the contraction of e from M is connected. If, for every element of M, exactly one of these minors is connected, then we call M minor-minimally-connected. This paper characterizes such matroids and shows that they must contain a number of two-element circuits or cocircuits. In addition, a new bound is proved on the number of 2-cocircuits in a minimally connected matroid

Matroid toric ideals: complete intersection, minors and minimal systems of generators

In this paper, we investigate three problems concerning the toric ideal associated to a matroid. Firstly, we list all matroids $\mathcal M$ such that its corresponding toric ideal $I_{\mathcal M}$ is a complete intersection. Secondly, we handle with the problem of detecting minors of a matroid $\mathcal M$ from a minimal set of binomial generators of $I_{\mathcal M}$. In particular, given a minimal set of binomial generators of $I_{\mathcal M}$ we provide a necessary condition for $\mathcal M$ to have a minor isomorphic to $\mathcal U_{d,2d}$ for $d \geq 2$. This condition is proved to be sufficient for $d = 2$ (leading to a criterion for determining whether $\mathcal M$ is binary) and for $d = 3$. Finally, we characterize all matroids $\mathcal M$ such that $I_{\mathcal M}$ has a unique minimal set of binomial generators.Comment: 9 page

Natural realizations of sparsity matroids

A hypergraph G with n vertices and m hyperedges with d endpoints each is (k,l)-sparse if for all sub-hypergraphs G' on n' vertices and m' edges, m'\le kn'-l. For integers k and l satisfying 0\le l\le dk-1, this is known to be a linearly representable matroidal family. Motivated by problems in rigidity theory, we give a new linear representation theorem for the (k,l)-sparse hypergraphs that is natural; i.e., the representing matrix captures the vertex-edge incidence structure of the underlying hypergraph G.Comment: Corrected some typos from the previous version; to appear in Ars Mathematica Contemporane

Slider-pinning Rigidity: a Maxwell-Laman-type Theorem

We define and study slider-pinning rigidity, giving a complete combinatorial characterization. This is done via direction-slider networks, which are a generalization of Whiteley's direction networks.Comment: Accepted, to appear in Discrete and Computational Geometr

Displaying blocking pairs in signed graphs

A signed graph is a pair (G, S) where G is a graph and S is a subset of the edges of G. A circuit of G is even (resp. odd) if it contains an even (resp. odd) number of edges of S. A blocking pair of (G, S) is a pair of vertices s, t such that every odd circuit intersects at least one of s or t. In this paper, we characterize when the blocking pairs of a signed graph can be represented by 2-cuts in an auxiliary graph. We discuss the relevance of this result to the problem of recognizing even cycle matroids and to the problem of characterizing signed graphs with no odd-K5 minor

The Family of Bicircular Matroids Closed Under Duality

We characterize the 3-connected members of the intersection of the class of bicircular and cobi- circular matroids. Aside from some exceptional matroids with rank and corank at most 5, this class consists of just the free swirls and their minors