We specify what is meant for a polytope to be reconstructible from its graph or dual graph, and we introduce the problem of class reconstructibility; i.e., face lattice can determined (dual) within given class. provide examples cubical polytopes that are not their graphs. Furthermore, show matroid (base) graphs graphs; our counterexamples include hypersimplices. Additionally, prove graphs, pres...

A Rayleigh matroid is one which satisfies a set of inequalities analogous to the Rayleigh monotonicity property of linear resistive electrical networks. We show that every matroid of rank three satisfies these inequalities.

We discuss the relationship between the vertical connectivity of a biased graph Ω and the Tutte connectivity of the frame matroid of Ω (also known as the bias matroid of Ω).

In this note we investigate some matroid minor structure results. In particular, we present sufficient conditions, in terms of triangles, for a matroid to have either U2,4 or F7 or M(K5) as a minor.

We prove that the maximum size of a simple binary matroid of rank r ≥ 5 with no AG(3, 2)-minor is (r+1 2 ) and characterize those matroids achieving this bound. When r ≥ 6, the graphic matroid M(Kr+1) is the unique matroid meeting the bound, but there are a handful of matroids of lower ranks meeting or exceeding this bound. In addition, we determine the size function for nongraphic simple binar...

A sufficiently large connected matroid M contains a big circuit or a big cocircuit. Pou-Lin Wu showed that we can ensure that M has a big circuit or a big cocircuit containing any chosen element of M . In this paper, we begin with a fixed connected matroid N and we take M to be a connected matroid that has N as a minor. Our main result establishes that ifM is sufficiently large, then, up to dua...

Rough sets are efficient for data pre-processing in data mining. As a generalization of the linear independence in vector spaces, matroids provide wellestablished platforms for greedy algorithms. In this paper, we apply rough sets to matroids and study the contraction of the dual of the corresponding matroid. First, for an equivalence relation on a universe, a matroidal structure of the rough s...

We derive a general result concerning cyclic orderings of the elements of a matroid. As corollaries we obtain two further results. The first corollary proves a conjecture of Gonçalves [7], stating that the circular arboricity of a matroid is equal to its fractional arboricity. This generalises a well-known result from Nash-Williams on covering graphs by spanning trees, and a result from Edmonds...

This article introduces the lazy matroid problem, which captures the goal of saving time or money in certain task selection scenarios. We are given a budget B and a matroid M with weights on its elements. The problem consists in finding an independent set F of minimum weight. In addition, F is feasible if its augmentation with any new element x implies that either F + x exceeds B or F + x is de...

Motivated by a property of linear resistive electrical networks, we introduce the class of Rayleigh matroids. These form a subclass of the balanced matroids defined by Feder and Mihail [10] in 1992. We prove a variety of results relating Rayleigh matroids to other well–known classes – in particular, we show that a binary matroid is Rayleigh if and only if it does not contain S8 as a minor. This...