6 research outputs found
The universality theorem for neighborly polytopes
In this note, we prove that every open primary basic semialgebraic set is
stably equivalent to the realization space of an even-dimensional neighborly
polytope. This in particular provides the final step for Mn\"ev's proof of the
universality theorem for simplicial polytopes.Comment: 5 pages, 1 figure. Small change
Universality theorems for inscribed polytopes and Delaunay triangulations
We prove that every primary basic semialgebraic set is homotopy equivalent to
the set of inscribed realizations (up to M\"obius transformation) of a
polytope. If the semialgebraic set is moreover open, then, in addition, we
prove that (up to homotopy) it is a retract of the realization space of some
inscribed neighborly (and simplicial) polytope. We also show that all algebraic
extensions of are needed to coordinatize inscribed polytopes.
These statements show that inscribed polytopes exhibit the Mn\"ev universality
phenomenon.
Via stereographic projections, these theorems have a direct translation to
universality theorems for Delaunay subdivisions. In particular, our results
imply that the realizability problem for Delaunay triangulations is
polynomially equivalent to the existential theory of the reals.Comment: 15 pages, 2 figure
Integer realizations of disk and segment graphs
A disk graph is the intersection graph of disks in the plane, a unit disk
graph is the intersection graph of same radius disks in the plane, and a
segment graph is an intersection graph of line segments in the plane. It can be
seen that every disk graph can be realized by disks with centers on the integer
grid and with integer radii; and similarly every unit disk graph can be
realized by disks with centers on the integer grid and equal (integer) radius;
and every segment graph can be realized by segments whose endpoints lie on the
integer grid. Here we show that there exist disk graphs on vertices such
that in every realization by integer disks at least one coordinate or radius is
and on the other hand every disk graph can be realized by
disks with integer coordinates and radii that are at most ; and
we show the analogous results for unit disk graphs and segment graphs. For
(unit) disk graphs this answers a question of Spinrad, and for segment graphs
this improves over a previous result by Kratochv\'{\i}l and Matou{\v{s}}ek.Comment: 35 pages, 14 figures, corrected a typ