186 research outputs found
The excluded minors for near-regular matroids
In unpublished work, Geelen proved that a matroid is near-regular if and only if it has no minor isomorphic to: U2,5; U3,5; the Fano plane and its dual; the non-Fano and its dual; the single-element deletion of AG(2,3), its dual, and the matroid obtained from it with a Delta-Y operation; and P8. We provide a proof of this characterization
Bicircular signed-graphic matroids
Several matroids can be defined on the edge set of a graph. Although
historically the cycle matroid has been the most studied, in recent times, the
bicircular matroid has cropped up in several places. A theorem of Matthews from
late 1970s gives a characterization of graphs whose bicircular matroids are
graphic. We give a characterization of graphs whose bicircular matroids are
signed-graphic.Comment: 8 page
Confinement of matroid representations to subsets of partial fields
Let M be a matroid representable over a (partial) field P and B a matrix
representable over a sub-partial field P' of P. We say that B confines M to P'
if, whenever a P-representation matrix A of M has a submatrix B, A is a scaled
P'-matrix. We show that, under some conditions on the partial fields, on M, and
on B, verifying whether B confines M to P' amounts to a finite check. A
corollary of this result is Whittle's Stabilizer Theorem.
A combination of the Confinement Theorem and the Lift Theorem from
arXiv:0804.3263 leads to a short proof of Whittle's characterization of the
matroids representable over GF(3) and other fields.
We also use a combination of the Confinement Theorem and the Lift Theorem to
prove a characterization, in terms of representability over partial fields, of
the 3-connected matroids that have k inequivalent representations over GF(5),
for k = 1, ..., 6.
Additionally we give, for a fixed matroid M, an algebraic construction of a
partial field P_M and a representation A over P_M such that every
representation of M over a partial field P is equal to f(A) for some
homomorphism f:P_M->P. Using the Confinement Theorem we prove an algebraic
analog of the theory of free expansions by Geelen et al.Comment: 45 page
An obstacle to a decomposition theorem for near-regular matroids
Seymour's Decomposition Theorem for regular matroids states that any matroid
representable over both GF(2) and GF(3) can be obtained from matroids that are
graphic, cographic, or isomorphic to R10 by 1-, 2-, and 3-sums. It is hoped
that similar characterizations hold for other classes of matroids, notably for
the class of near-regular matroids. Suppose that all near-regular matroids can
be obtained from matroids that belong to a few basic classes through k-sums.
Also suppose that these basic classes are such that, whenever a class contains
all graphic matroids, it does not contain all cographic matroids. We show that
in that case 3-sums will not suffice.Comment: 11 pages, 1 figur
Computing excluded minors for classes of matroids representable over partial fields
We describe an implementation of a computer search for the "small" excluded minors for a class of matroids representable over a partial field. Using these techniques, we enumerate the excluded minors on at most 15 elements for both the class of dyadic matroids, and the class of 2-regular matroids. We conjecture that there are no other excluded minors for the class of 2-regular matroids; whereas, on the other hand, we show that there is a 16-element excluded minor for the class of dyadic matroids.We describe an implementation of a computer search for the "small" excluded minors for a class of matroids representable over a partial field. Using these techniques, we enumerate the excluded minors on at most 15 elements for both the class of dyadic matroids, and the class of 2-regular matroids. We conjecture that there are no other excluded minors for the class of 2-regular matroids; whereas, on the other hand, we show that there is a 16-element excluded minor for the class of dyadic matroids
Representing some non-representable matroids
We extend the notion of representation of a matroid to algebraic structures
that we call skew partial fields. Our definition of such representations
extends Tutte's definition, using chain groups. We show how such
representations behave under duality and minors, we extend Tutte's
representability criterion to this new class, and we study the generator
matrices of the chain groups. An example shows that the class of matroids
representable over a skew partial field properly contains the class of matroids
representable over a skew field.
Next, we show that every multilinear representation of a matroid can be seen
as a representation over a skew partial field.
Finally we study a class of matroids called quaternionic unimodular. We prove
a generalization of the Matrix Tree theorem for this class.Comment: 29 pages, 2 figure
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