16 research outputs found
Graphical representations of graphic frame matroids
A frame matroid M is graphic if there is a graph G with cycle matroid
isomorphic to M. In general, if there is one such graph, there will be many.
Zaslavsky has shown that frame matroids are precisely those having a
representation as a biased graph; this class includes graphic matroids,
bicircular matroids, and Dowling geometries. Whitney characterized which graphs
have isomorphic cycle matroids, and Matthews characterised which graphs have
isomorphic graphic bicircular matroids. In this paper, we give a
characterization of which biased graphs give rise to isomorphic graphic frame
matroids
Defining bicircular matroids in monadic logic
We conjecture that the class of frame matroids can be characterised by a
sentence in the monadic second-order logic of matroids, and we prove that there
is such a characterisation for the class of bicircular matroids. The proof does
not depend on an excluded-minor characterisation
Even Cycle and Even Cut Matroids
In this thesis we consider two classes of binary matroids, even cycle matroids and even cut matroids. They are a generalization of graphic and cographic matroids respectively. We focus on two main problems for these classes of matroids. We first consider the Isomorphism Problem, that is the relation between two representations of the same matroid. A representation of an even cycle matroid is a pair formed by a graph together with a special set of edges of the graph. Such a pair is called a signed graph. A representation for an even cut matroid is a pair formed by a graph together with a special set of vertices of the graph. Such a pair is called a graft. We show that two signed graphs representing the same even cycle matroid relate to two grafts representing the same even cut matroid. We then present two classes of signed graphs and we solve the Isomorphism Problem for these two classes. We conjecture that any two representations of the same even cycle matroid are either in one of these two classes, or are related by a local modification of a known operation, or form a sporadic example. The second problem we consider is finding the excluded minors for these classes of matroids. A difficulty when looking for excluded minors for these classes arises from the fact that in general the matroids may have an arbitrarily large number of representations. We define degenerate even cycle and even cut matroids. We show that a 3-connected even cycle matroid containing a 3-connected non-degenerate minor has, up to a simple equivalence relation, at most twice as many representations as the minor. We strengthen this result for a particular class of non-degenerate even cycle matroids. We also prove analogous results for even cut matroids
Graphs with the same truncated cycle matroid
The classical Whitney's 2-Isomorphism Theorem describes the families of
graphs having the same cycle matroid. In this paper we describe the families of
graphs having the same truncated cycle matroid and prove, in particular, that
every 3-connected graph, except for K4, is uniquely defined by its truncated
cycle matroid.Comment: 8 page
Monadic transductions and definable classes of matroids
A transduction provides us with a way of using the monadic second-order
language of a structure to make statements about a derived structure. Any
transduction induces a relation on the set of these structures. This article
presents a self-contained presentation of the theory of transductions for the
monadic second-order language of matroids. This includes a proof of the matroid
version of the Backwards Translation Theorem, which lifts any formula applied
to the images of the transduction into a formula which we can apply to the
pre-images. Applications include proofs that the class of lattice-path matroids
and the class of spike-minors can be defined by sentences in monadic
second-order logic