31 research outputs found
Infinite Matroids and Determinacy of Games
Solving a problem of Diestel and Pott, we construct a large class of infinite
matroids. These can be used to provide counterexamples against the natural
extension of the Well-quasi-ordering-Conjecture to infinite matroids and to
show that the class of planar infinite matroids does not have a universal
matroid.
The existence of these matroids has a connection to Set Theory in that it
corresponds to the Determinacy of certain games. To show that our construction
gives matroids, we introduce a new very simple axiomatization of the class of
countable tame matroids
Fixing numbers for matroids
Motivated by work in graph theory, we define the fixing number for a matroid.
We give upper and lower bounds for fixing numbers for a general matroid in
terms of the size and maximum orbit size (under the action of the matroid
automorphism group). We prove the fixing numbers for the cycle matroid and
bicircular matroid associated with 3-connected graphs are identical. Many of
these results have interpretations through permutation groups, and we make this
connection explicit.Comment: This is a major revision of a previous versio
Totally free expansions of matroids
The aim of this paper is to give insight into the behaviour of inequivalent representations of 3-connected matroids. An element x of a matroid M is fixed if there is no extension M′ of M by an element x′ such that {x, x′} is independent and M′ is unaltered by swapping the labels on x and x′. When x is fixed, a representation of M.\x extends in at most one way to a representation of M. A 3-connected matroid N is totally free if neither N nor its dual has a fixed element whose deletion is a series extension of a 3-connected matroid. The significance of such matroids derives from the theorem, established here, that the number of inequivalent representations of a 3-connected matroid M over a finite field F is bounded above by the maximum, over all totally free minors N of M, of the number of inequivalent F-representations of N. It is proved that, within a class of matroids that is closed under minors and duality, the totally free matroids can be found by an inductive search. Such a search is employed to show that, for all r ≥ 4, there are unique and easily described rank-r quaternary and quinternary matroids, the first being the free spike. Finally, Seymour\u27s Splitter Theorem is extended by showing that the sequence of 3-connected matroids from a matroid M to a minor N, whose existence is guaranteed by the theorem, may be chosen so that all deletions and contractions of fixed and cofixed elements occur in the initial segment of the sequence. © 2001 Elsevier Science
Representative set statements for delta-matroids and the Mader delta-matroid
We present representative sets-style statements for linear delta-matroids,
which are set systems that generalize matroids, with important connections to
matching theory and graph embeddings. Furthermore, our proof uses a new
approach of sieving polynomial families, which generalizes the linear algebra
approach of the representative sets lemma to a setting of bounded-degree
polynomials. The representative sets statements for linear delta-matroids then
follow by analyzing the Pfaffian of the skew-symmetric matrix representing the
delta-matroid. Applying the same framework to the determinant instead of the
Pfaffian recovers the representative sets lemma for linear matroids.
Altogether, this significantly extends the toolbox available for kernelization.
As an application, we show an exact sparsification result for Mader networks:
Let be a graph and a partition of a set of terminals , . A -path in is a path with endpoints
in distinct parts of and internal vertices disjoint from . In
polynomial time, we can derive a graph with ,
such that for every subset there is a packing of
-paths with endpoints in if and only if there is one in
, and . This generalizes the (undirected version of the)
cut-covering lemma, which corresponds to the case that contains
only two blocks.
To prove the Mader network sparsification result, we furthermore define the
class of Mader delta-matroids, and show that they have linear representations.
This should be of independent interest
Inequivalent Representations of Matroids over Prime Fields
It is proved that for each prime field , there is an integer
such that a 4-connected matroid has at most inequivalent representations
over . We also prove a stronger theorem that obtains the same conclusion
for matroids satisfying a connectivity condition, intermediate between
3-connectivity and 4-connectivity that we term "-coherence".
We obtain a variety of other results on inequivalent representations
including the following curious one. For a prime power , let denote the set of matroids representable over all fields with at least
elements. Then there are infinitely many Mersenne primes if and only if,
for each prime power , there is an integer such that a 3-connected
member of has at most inequivalent
GF(7)-representations.
The theorems on inequivalent representations of matroids are consequences of
structural results that do not rely on representability. The bulk of this paper
is devoted to proving such results
The Linkage Problem for Group-labelled Graphs
This thesis aims to extend some of the results of the Graph Minors Project of Robertson and Seymour to "group-labelled graphs". Let be a group. A -labelled graph is an oriented graph with its edges labelled from , and is thus a generalization of a signed graph.
Our primary result is a generalization of the main result from Graph Minors XIII. For any finite abelian group , and any fixed -labelled graph , we present a polynomial-time algorithm that determines if an input -labelled graph has an -minor. The correctness of our algorithm relies on much of the machinery developed throughout the graph minors papers. We therefore hope it can serve as a reasonable introduction to the subject.
Remarkably, Robertson and Seymour also prove that for any sequence of graphs, there exist indices such that is isomorphic to a minor of . Geelen, Gerards and Whittle recently announced a proof of the analogous result for -labelled graphs, for finite abelian. Together with the main result of this thesis, this implies that membership in any minor closed class of -labelled graphs can be decided in polynomial-time. This also has some implications for well-quasi-ordering certain classes of matroids, which we discuss