1,460 research outputs found
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
Euclidean distance geometry and applications
Euclidean distance geometry is the study of Euclidean geometry based on the
concept of distance. This is useful in several applications where the input
data consists of an incomplete set of distances, and the output is a set of
points in Euclidean space that realizes the given distances. We survey some of
the theory of Euclidean distance geometry and some of the most important
applications: molecular conformation, localization of sensor networks and
statics.Comment: 64 pages, 21 figure
Witness (Delaunay) Graphs
Proximity graphs are used in several areas in which a neighborliness
relationship for input data sets is a useful tool in their analysis, and have
also received substantial attention from the graph drawing community, as they
are a natural way of implicitly representing graphs. However, as a tool for
graph representation, proximity graphs have some limitations that may be
overcome with suitable generalizations. We introduce a generalization, witness
graphs, that encompasses both the goal of more power and flexibility for graph
drawing issues and a wider spectrum for neighborhood analysis. We study in
detail two concrete examples, both related to Delaunay graphs, and consider as
well some problems on stabbing geometric objects and point set discrimination,
that can be naturally described in terms of witness graphs.Comment: 27 pages. JCCGG 200
Recognition and Complexity of Point Visibility Graphs
A point visibility graph is a graph induced by a set of points in the plane, where every vertex corresponds to a point, and two vertices are adjacent whenever the two corresponding points are visible from each other, that is, the open segment between them does not contain any other point of the set.
We study the recognition problem for point visibility graphs: given a simple undirected graph, decide whether it is the visibility graph of some point set in the plane. We show that the problem is complete for the existential theory of the reals. Hence the problem is as hard as deciding the existence of a real solution to a system of polynomial inequalities. The proof involves simple substructures forcing collinearities in all realizations of some visibility graphs, which are applied to the algebraic universality constructions of Mnev and Richter-Gebert. This solves a longstanding open question and paves the way for the analysis of other classes of visibility graphs.
Furthermore, as a corollary of one of our construction, we show that there exist point visibility graphs that do not admit any geometric realization with points having integer coordinates
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