569 research outputs found
First order convergence of matroids
The model theory based notion of the first order convergence unifies the
notions of the left-convergence for dense structures and the Benjamini-Schramm
convergence for sparse structures. It is known that every first order
convergent sequence of graphs with bounded tree-depth can be represented by an
analytic limit object called a limit modeling. We establish the matroid
counterpart of this result: every first order convergent sequence of matroids
with bounded branch-depth representable over a fixed finite field has a limit
modeling, i.e., there exists an infinite matroid with the elements forming a
probability space that has asymptotically the same first order properties. We
show that neither of the bounded branch-depth assumption nor the
representability assumption can be removed.Comment: Accepted to the European Journal of Combinatoric
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
Fork-decompositions of matroids
For the abstract of this paper, please see the PDF file
Branch-depth: Generalizing tree-depth of graphs
We present a concept called the branch-depth of a connectivity function, that
generalizes the tree-depth of graphs. Then we prove two theorems showing that
this concept aligns closely with the notions of tree-depth and shrub-depth of
graphs as follows. For a graph and a subset of we let
be the number of vertices incident with an edge in and an
edge in . For a subset of , let be the rank
of the adjacency matrix between and over the binary field.
We prove that a class of graphs has bounded tree-depth if and only if the
corresponding class of functions has bounded branch-depth and
similarly a class of graphs has bounded shrub-depth if and only if the
corresponding class of functions has bounded branch-depth, which we
call the rank-depth of graphs.
Furthermore we investigate various potential generalizations of tree-depth to
matroids and prove that matroids representable over a fixed finite field having
no large circuits are well-quasi-ordered by the restriction.Comment: 34 pages, 2 figure
The matroid secretary problem for minor-closed classes and random matroids
We prove that for every proper minor-closed class of matroids
representable over a prime field, there exists a constant-competitive matroid
secretary algorithm for the matroids in . This result relies on the
extremely powerful matroid minor structure theory being developed by Geelen,
Gerards and Whittle.
We also note that for asymptotically almost all matroids, the matroid
secretary algorithm that selects a random basis, ignoring weights, is
-competitive. In fact, assuming the conjecture that almost all
matroids are paving, there is a -competitive algorithm for almost all
matroids.Comment: 15 pages, 0 figure
Branch-depth: Generalizing tree-depth of graphs
We present a concept called the branch-depth of a connectivity function, that
generalizes the tree-depth of graphs. Then we prove two theorems showing that
this concept aligns closely with the notions of tree-depth and shrub-depth of
graphs as follows. For a graph and a subset of we let
be the number of vertices incident with an edge in and an
edge in . For a subset of , let be the rank
of the adjacency matrix between and over the binary field.
We prove that a class of graphs has bounded tree-depth if and only if the
corresponding class of functions has bounded branch-depth and
similarly a class of graphs has bounded shrub-depth if and only if the
corresponding class of functions has bounded branch-depth, which we
call the rank-depth of graphs.
Furthermore we investigate various potential generalizations of tree-depth to
matroids and prove that matroids representable over a fixed finite field having
no large circuits are well-quasi-ordered by the restriction.Comment: 36 pages, 2 figures. Final versio
On matroids of branch-width three
For the abstract of this paper, please see the PDF file
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