520 research outputs found
There is no tame triangulation of the infinite real Grassmannian
We show that there is no triangulation of the infinite real Grassmannian of
k-planes in R^\infty which is nicely situated with respect to the coordinate
axes. In terms of matroid theory, this says there is no triangulation of the
Grassmannian subdividing the matroid stratification. This is proved by an
argument in projective geometry, considering a specific sequence of
configurations of points in the plane.Comment: 11 page
Convexity in partial cubes: the hull number
We prove that the combinatorial optimization problem of determining the hull
number of a partial cube is NP-complete. This makes partial cubes the minimal
graph class for which NP-completeness of this problem is known and improves
some earlier results in the literature.
On the other hand we provide a polynomial-time algorithm to determine the
hull number of planar partial cube quadrangulations.
Instances of the hull number problem for partial cubes described include
poset dimension and hitting sets for interiors of curves in the plane.
To obtain the above results, we investigate convexity in partial cubes and
characterize these graphs in terms of their lattice of convex subgraphs,
improving a theorem of Handa. Furthermore we provide a topological
representation theorem for planar partial cubes, generalizing a result of
Fukuda and Handa about rank three oriented matroids.Comment: 19 pages, 4 figure
Tropical types and associated cellular resolutions
An arrangement of finitely many tropical hyperplanes in the tropical torus
leads to a notion of `type' data for points, with the underlying unlabeled
arrangement giving rise to `coarse type'. It is shown that the decomposition of
the tropical torus induced by types gives rise to minimal cocellular
resolutions of certain associated monomial ideals. Via the Cayley trick from
geometric combinatorics this also yields cellular resolutions supported on
mixed subdivisions of dilated simplices, extending previously known
constructions. Moreover, the methods developed lead to an algebraic algorithm
for computing the facial structure of arbitrary tropical complexes from point
data.Comment: minor correction
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