366 research outputs found
Quasi-graphic matroids
Frame matroids and lifted-graphic matroids are two interesting generalizations of graphic matroids. Here we introduce a new generalization, quasi-graphic matroids, that unifies these two existing classes. Unlike frame matroids and lifted-graphic matroids, it is easy to certify that a matroid is quasi-graphic. The main result of the paper is that every 3-connected representable quasi-graphic matroid is either a lifted-graphic matroid or a rame matroid
Describing Quasi-Graphic Matroids
The class of quasi-graphic matroids recently introduced by Geelen, Gerards,
and Whittle generalises each of the classes of frame matroids and
lifted-graphic matroids introduced earlier by Zaslavsky. For each biased graph
Zaslavsky defined a unique lift matroid
and a unique frame matroid , each on ground set . We
show that in general there may be many quasi-graphic matroids on and
describe them all. We provide cryptomorphic descriptions in terms of subgraphs
corresponding to circuits, cocircuits, independent sets, and bases. Equipped
with these descriptions, we prove some results about quasi-graphic matroids. In
particular, we provide alternate proofs that do not require 3-connectivity of
two results of Geelen, Gerards, and Whittle for 3-connected matroids from their
introductory paper: namely, that every quasi-graphic matroid linearly
representable over a field is either lifted-graphic or frame, and that if a
matroid has a framework with a loop that is not a loop of then is
either lifted-graphic or frame. We also provide sufficient conditions for a
quasi-graphic matroid to have a unique framework.
Zaslavsky has asked for those matroids whose independent sets are contained
in the collection of independent sets of while containing
those of , for some biased graph . Adding a
natural (and necessary) non-degeneracy condition defines a class of matroids,
which we call biased graphic. We show that the class of biased graphic matroids
almost coincides with the class of quasi-graphic matroids: every quasi-graphic
matroid is biased graphic, and if is a biased graphic matroid that is not
quasi-graphic then is a 2-sum of a frame matroid with one or more
lifted-graphic matroids
The -connected Excluded Minors for the Class of Quasi-graphic Matroids
The class of quasi-graphic matroids, recently introduced by Geelen, Gerards,
and Whittle, is minor closed and contains both the class of lifted-graphic
matroids and the class of frame matroids, each of which generalises the class
of graphic matroids. In this paper, we prove that the matroids and
are the only -connected excluded minors for the class of
quasi-graphic matroids
There are only a finite number of excluded minors for the class of bicircular matroids
We show that the class of bicircular matroids has only a finite number of
excluded minors. Key tools used in our proof include representations of
matroids by biased graphs and the recently introduced class of quasi-graphic
matroids. We show that if is an excluded minor of rank at least ten, then
is quasi-graphic. Several small excluded minors are quasi-graphic. Using
biased-graphic representations, we find that already contains one of these.
We also provide an upper bound, in terms of rank, on the number of elements in
an excluded minor, so the result follows.Comment: Added an appendix describing all known excluded minors. Added Gordon
Royle as author. Some proofs revised and correcte
Dessins, their delta-matroids and partial duals
Given a map on a connected and closed orientable surface, the
delta-matroid of is a combinatorial object associated to which captures some topological information of the embedding. We explore how
delta-matroids associated to dessins d'enfants behave under the action of the
absolute Galois group. Twists of delta-matroids are considered as well; they
correspond to the recently introduced operation of partial duality of maps.
Furthermore, we prove that every map has a partial dual defined over its field
of moduli. A relationship between dessins, partial duals and tropical curves
arising from the cartography groups of dessins is observed as well.Comment: 34 pages, 20 figures. Accepted for publication in the SIGMAP14
Conference Proceeding
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