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DESCRIPTION: Towards a better understanding of arrangements of circles and
also to get rid of geometric difficulties\, we look at the more general set
ting of ''arrangements of pseudocircles'' which was first introduced by Grü
nbaum in the 1970's. An arrangement of pseudocircles is a collection of sim
ple closed curves on the sphere or in the plane such that any two of the cu
rves are either disjoint or intersect in exactly two points\, where the two
curves cross. In his book\, Grünbaum conjectured that every digon-free arr
angement of n pairwise intersecting pseudocircles contains at least $2n-4$
triangular cells. We present arrangements to disprove this conjecture and g
ive new bounds on the number of triangular cells for various classes of arr
angements. Furthermore\, we study the ''circularizability'' of arrangemen
ts: it is clear that every arrangement of circles is an arrangement of pseu
docircles\, however\, deciding whether an arrangement of pseudocircles is i
somorphic to an arrangement of circles is computationally hard. Using a com
puter program\, we have enumerated all combinatorially different arrangemen
ts of up to $7$ pseudocircles. For the class of arrangements of $5$ pseudoc
ircles and for the class of digon-free intersecting arrangements of $6$ pse
udocircles\, we give a complete classification: we either provide a circle
representation or a non-circularizability proof. For these proofs we use in
cidence theorems like Miquel's and arguments based on continuous deformatio
n\, where circles of an assumed circle representation grow or shrink in a c
ontrolled way. This talk summarizes results from two articles\, which are
both joint work with Stefan Felsner: * Arrangements of Pseudocircles: Tria
ngles and Drawings\; short version in Proc. GD'17\; full version available
at arXiv (1708.06449) * Arrangements of Pseudocircles: On Circularizability
\; short version in Proc. GD'18\; full version in DCG: Ricky Pollack Memori
al Issue (doi:10.1007/s00454-019-00077-y)
DTSTAMP:20190524T131600
DTSTART:20190701T160000
CLASS:PUBLIC
LOCATION:Freie Universität Berlin \n Institut für Informatik \n Takustr. 9
\n 14195 Berlin \n Room 005 (Ground Floor)
SEQUENCE:0
SUMMARY:Manfred Scheucher (Technische Universität Berlin): On Arrangements
of Pseudocircles
UID:95691268@www.facetsofcomplexity.de
URL:http://www.facetsofcomplexity.de/monday/20190701-C-Scheucher.html
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