> circles
Go back...
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).
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:
We provide realizations for all biconnected arrangements
of $n=5$ pseudocircles:
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.
Go back...Last update: July 09 2022 12:31:27.
(c) 2017 Stefan Felsner and Manfred Scheucher