294 research outputs found
Algebraic matroids with graph symmetry
This paper studies the properties of two kinds of matroids: (a) algebraic
matroids and (b) finite and infinite matroids whose ground set have some
canonical symmetry, for example row and column symmetry and transposition
symmetry.
For (a) algebraic matroids, we expose cryptomorphisms making them accessible
to techniques from commutative algebra. This allows us to introduce for each
circuit in an algebraic matroid an invariant called circuit polynomial,
generalizing the minimal poly- nomial in classical Galois theory, and studying
the matroid structure with multivariate methods.
For (b) matroids with symmetries we introduce combinatorial invariants
capturing structural properties of the rank function and its limit behavior,
and obtain proofs which are purely combinatorial and do not assume algebraicity
of the matroid; these imply and generalize known results in some specific cases
where the matroid is also algebraic. These results are motivated by, and
readily applicable to framework rigidity, low-rank matrix completion and
determinantal varieties, which lie in the intersection of (a) and (b) where
additional results can be derived. We study the corresponding matroids and
their associated invariants, and for selected cases, we characterize the
matroidal structure and the circuit polynomials completely
Algebraic matroids with graph symmetry
This paper studies the properties of two kinds of matroids: (a) algebraic matroids
and (b) finite and infinite matroids whose ground set have some canonical symmetry,
for example row and column symmetry and transposition symmetry.
For (a) algebraic matroids, we expose cryptomorphisms making them accessible to
techniques from commutative algebra. This allows us to introduce for each circuit in an
algebraic matroid an invariant called circuit polynomial, generalizing the minimal polynomial
in classical Galois theory, and studying the matroid structure with multivariate
methods.
For (b) matroids with symmetries we introduce combinatorial invariants capturing
structural properties of the rank function and its limit behavior, and obtain proofs which
are purely combinatorial and do not assume algebraicity of the matroid; these imply
and generalize known results in some specific cases where the matroid is also algebraic.
These results are motivated by, and readily applicable to framework rigidity, low-rank
matrix completion and determinantal varieties, which lie in the intersection of (a) and (b)
where additional results can be derived. We study the corresponding matroids and their
associated invariants, and for selected cases, we characterize the matroidal structure
and the circuit polynomials completely
A tight relation between series--parallel graphs and bipartite distance hereditary graphs
Bandelt and Mulder’s structural characterization of bipartite distance hereditary graphs asserts that such graphs can be built inductively starting from a single vertex and by re17 peatedly adding either pendant vertices or twins (i.e., vertices with the same neighborhood as an existing one). Dirac and Duffin’s structural characterization of 2–connected series–parallel graphs asserts that such graphs can be built inductively starting from a single edge by adding either edges in series or in parallel. In this paper we give an elementary proof that the two constructions are the same construction when bipartite graphs are viewed as the fundamental graphs of a graphic matroid. We then apply the result to re-prove known results concerning bipartite distance hereditary graphs and series–parallel graphs and to provide a new class of polynomially-solvable instances for the integer multi-commodity flow of maximum valu
Tropical varieties, maps and gossip
Tropical geometry is a relatively new field of mathematics that studies the tropicalization map: a map that assigns a certain type of polyhedral complex, called a tropical variety, to an embedded algebraic variety. In a sense, it translates algebraic geometric statements into combinatorial ones. An interesting feature of tropical geometry is that there does not exist a good notion of morphism, or map, between tropical varieties that makes the tropicalization map functorial. The main part of this thesis studies maps between different classes of tropical varieties: tropical linear spaces and tropicalizations of embedded unirational varieties. The first chapter is a concise introduction to tropical geometry. It collects and proves the main theorems. None of these results are new. The second chapter deals with tropicalizations of embedded unirational varieties. We give sufficient conditions on such varieties for there to exist a (not necessarily injective) parametrization whose naive tropicalization is surjective onto the associated tropical variety. The third chapter gives an overview of the algebra related to tropical linear spaces. Where fields and vector spaces are the central objects in linear algebra, so are semifields and modules over semifields central to tropical linear algebra and the study of tropical linear spaces. Most results in this chapter are known in some form, but scattered among the available literature. The main purpose of this chapter is to collect these results and to determine the algebraic conditions that suffice to give linear algebra over the semifield a familiar feel. For example, under which conditions are varieties cut out by linear polynomials closed under addition and scalar multiplication? The fourth chapter comprises the biggest part of the thesis. The techniques used are a combination of tropical linear algebra and matroid theory. Central objects are the valuated matroids introduced by Andreas Dress and Walter Wenzl. Among other things the chapter contains a classification of functions on a tropical linear space whose cycles are tropical linear subspaces, extending an old result on elementary extensions of matroids by Henry Crapo. It uses Mikhalkin’s concept of a tropical modification to define the morphisms in a category whose objects are all tropical linear spaces. Finally, we determine the structure of an open submonoid of the morphisms from affine 2-space to itself as a polyhedral complex. Finally, the fifth and last chapter is only indirectly related to maps. It studies a certain monoid contained in the tropicalization of the orthogonal group: the monoid that is generated by the distance matrices under tropical matrix multiplication (i.e. where addition is replaced by minimum, and multiplication by addition). This monoid generalizes a monoid that underlies the well-known gossip problem, to a setting where information is transmitted only with a certain degree accuracy. We determine this so-called gossip monoid for matrices up to size 4, and prove that in general it is a polyhedral monoid of dimension equal to that of the orthogonal group
The world of hereditary graph classes viewed through Truemper configurations
In 1982 Truemper gave a theorem that characterizes graphs whose edges can be labeled so that all chordless cycles have prescribed parities. The characterization states that this can be done for a graph G if and only if it can be done for all induced subgraphs of G that are of few speci c types, that we will call Truemper con gurations. Truemper was originally motivated by the problem of obtaining a co-NP characterization of bipartite graphs that are signable to be balanced (i.e. bipartite graphs whose node-node incidence matrices are balanceable matrices). The con gurations that Truemper identi ed in his theorem ended up playing a key role in understanding the structure of several seemingly diverse classes of objects, such as regular matroids, balanceable matrices and perfect graphs. In this survey we view all these classes, and more, through the excluded Truemper con gurations, focusing on the algorithmic consequences, trying to understand what structurally enables e cient recognition and optimization algorithms
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