Circuit Covers of Signed Eulerian Graphs

A signed circuit cover of a signed graph is a natural analog of a circuit cover of a graph, and is equivalent to a covering of its corresponding signed-graphic matroid with circuits. It was conjectured that a signed graph whose signed-graphic matroid has no coloops has a 6-cover. In this paper, we prove that the conjecture holds for signed Eulerian graphs.

MODULI SPACES OF RATIONAL WEIGHTED STABLE CURVES AND TROPICAL GEOMETRY

We study moduli spaces of rational weighted stable tropical curves, and their connections with Hassett spaces. Given a vector $w$ of weights, the moduli space of tropical $w$-stable curves can be given the structure of a balanced fan if and only if $w$ has only heavy and light entries. In this case, the tropical moduli space can be expressed as the Bergman fan of an explicit graphic matroid. The tropical moduli space can be realized as a geometric tropicalization, and as a Berkovich skeleton, its algebraic counterpart. This builds on previous work of Tevelev, Gibney and Maclagan, and Abramovich, Caporaso and Payne. Finally, we construct the moduli spaces of heavy/light weighted tropical curves as fibre products of unweighted spaces, and explore parallels with the algebraic world.

Parallel algorithms for matroid intersection and matroid parity

A maximum linear matroid parity set is called a basic matroid parity set, if its size is the rank of the matroid. We show that determining the existence of a common base (basic matroid parity set) for linear matroid intersection (linear matroid parity) is in NC2, provided that there are polynomial number of common bases (basic matroid parity sets). For graphic matroids, we show that finding a common base for matroid intersection is in NC2, if the number of common bases is polynomial bounded. To our knowledge, these algorithms are the first deterministic NC algorithms for matroid intersection and matroid parity. We also give a new RNC2 algorithm that finds a common base for graphic matroid intersection. We prove that if there is a black-box NC algorithm for Polynomial Identity Testing (PIT), then there is an NC algorithm to determine the existence of a common base (basic matroid parity set) for linear matroid intersection (linear matroid parity).

Covering Cycle Matroid

Covering is a type of widespread data representation while covering-based rough sets provide an efficient and systematic theory to deal with this type of data. Matroids are based on linear algebra and graph theory and have a variety of applications in many fields. In this paper, we construct two types of covering cycle matroids by a covering and then study the graphical representations of these two types of matriods. First, through defining a cycle graph by a set, the type-1 covering cycle matroid is constructed by a covering. By a dual graph of the cycle graph, the covering can also induce the type-2 covering cycle matroid. Second, some characteristics of these two types of matroids are formulated by a covering, such as independent sets, bases, circuits, and support sets. Third, a coarse covering of a covering is defined to study the graphical representation of the type-1 covering cycle matroid. We prove that the type-1 covering cycle matroid is graphic while the type-2 covering cycle matroid is not always a graphic matroid. Finally, relationships between these two types of matroids and the function matroid are studied. In a word, borrowing from matroids, this work presents an interesting view, graph, to investigate covering-based rough sets.

Matroid matching with Dilworth truncation

International audience Let $H=(V,E)$ be a hypergraph and let $k≥ 1$ and$ l≥ 0$ be fixed integers. Let $\mathcal{M}$ be the matroid with ground-set $E s.t. a$ set $F⊆E$ is independent if and only if each $X⊆V$ with $k|X|-l≥ 0$ spans at most $k|X|-l$ hyperedges of $F$. We prove that if $H$ is dense enough, then $\mathcal{M}$ satisfies the double circuit property, thus the min-max formula of Dress and Lovász on the maximum matroid matching holds for $\mathcal{M}$ . Our result implies the Berge-Tutte formula on the maximum matching of graphs $(k=1, l=0)$, generalizes Lovász' graphic matroid (cycle matroid) matching formula to hypergraphs $(k=l=1)$ and gives a min-max formula for the maximum matroid matching in the 2-dimensional rigidity matroid $(k=2, l=3)$.