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On this page we provide circle representations for arrangements of pseudocircles. The files are in text format and each line contains a tuple, consisting of
• the sparse6-string of the graph corresponding to the arrangement and
• a list of circles, each described by a triple (x,y,r) -- actually ((x,y),r) -- where (x,y) describes the center and r describes the radius of the circle.
Some of the "radius" values r in our data come with a negative sign (we describe circles via the equation $x^2+y^2 = r^2$ and only care about the absolute values of x,y,r).

## Intersecting arrangements

For $n=4$ we provide realizations for all arrangements of pseudocircles. For $n=5$ we provide realizations for all 277 realizable intersecting arrangements (there are 278 intersecting arrangements and one of which is provable not circularizable). For $n=6$ we provide realizations for all 3128 realizable intersecting digonfree arrangements (there are 3131 intersecting digonfree arrangements and three of which are proveable not circularizable).
File: circles/intersecting/arr4_8_of_8.txt (663 bytes)

File: circles/intersecting/arr5_277_of_278.txt (32.91 KB)

File: circles/intersecting/arr6_digonfree_2128_of_2131.txt (324.50 KB)

## Biconnected arrangements

It is easy to see, that circles corresponding to cut-vertices of the intersection graph of a connected arrangement can be easily inserted (iteratively) after all biconnected components of the graph are realized. In contrary, every minimal non-circularizable arrangement of pseudocircles has a biconnected intersection graph. Therefore, we provide circle representations for "biconnected" arrangements. Note that the intersection graph of an intersecting arrangement is the complete graph.

Since arrangements with a biconnected intersection graph admit a dual graph with no multiple edges, we represent them using "model 1"; see this page. The database of biconnected arrangements of $n=5$ (model 1 encoded) is available here:
File: pseudocircles/biconnected/all_bicon_5.mod1s6 (32.62 KB)

We provide realizations for all biconnected arrangements of $n=5$ pseudocircles:
File: circles/biconnected/bicon_5_all.txt (161.31 KB)

## Further results

Besides the provided realizations, we by now also have realized arrangements
• about 4 400 connected digonfree arrangements of 6 circles (which is about 98%),
• about 130 000 intersecting arrangements of 6 circles (which is about 90%), and
• about 2 millions intersecting digonfree arrangements of 7 circles (which is about 66%).
These files are available on demand.

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Last update: October 29 2019 20:54:45. (c) 2017 Stefan Felsner and Manfred Scheucher