3,748 research outputs found
Construction of harmonic diffeomorphisms and minimal graphs
We study complete minimal graphs in HxR, which take asymptotic boundary
values plus and minus infinity on alternating sides of an ideal inscribed
polygon Γ in H. We give necessary and sufficient conditions on the
"lenghts" of the sides of the polygon (and all inscribed polygons in Γ)
that ensure the existence of such a graph. We then apply this to construct
entire minimal graphs in HxR that are conformally the complex plane C. The
vertical projection of such a graph yields a harmonic diffeomorphism from C
onto H, disproving a conjecture of Rick Schoen
Holes or Empty Pseudo-Triangles in Planar Point Sets
Let denote the smallest integer such that any set of at least
points in the plane, no three on a line, contains either an empty
convex polygon with vertices or an empty pseudo-triangle with
vertices. The existence of for positive integers ,
is the consequence of a result proved by Valtr [Discrete and Computational
Geometry, Vol. 37, 565--576, 2007]. In this paper, following a series of new
results about the existence of empty pseudo-triangles in point sets with
triangular convex hulls, we determine the exact values of and , and prove bounds on and , for . By
dropping the emptiness condition, we define another related quantity , which is the smallest integer such that any set of at least points in the plane, no three on a line, contains a convex polygon with
vertices or a pseudo-triangle with vertices. Extending a result of
Bisztriczky and T\'oth [Discrete Geometry, Marcel Dekker, 49--58, 2003], we
obtain the exact values of and , and obtain non-trivial
bounds on .Comment: A minor error in the proof of Theorem 2 fixed. Typos corrected. 19
pages, 11 figure
Graphs with Plane Outside-Obstacle Representations
An \emph{obstacle representation} of a graph consists of a set of polygonal
obstacles and a distinct point for each vertex such that two points see each
other if and only if the corresponding vertices are adjacent. Obstacle
representations are a recent generalization of classical polygon--vertex
visibility graphs, for which the characterization and recognition problems are
long-standing open questions.
In this paper, we study \emph{plane outside-obstacle representations}, where
all obstacles lie in the unbounded face of the representation and no two
visibility segments cross. We give a combinatorial characterization of the
biconnected graphs that admit such a representation. Based on this
characterization, we present a simple linear-time recognition algorithm for
these graphs. As a side result, we show that the plane vertex--polygon
visibility graphs are exactly the maximal outerplanar graphs and that every
chordal outerplanar graph has an outside-obstacle representation.Comment: 12 pages, 7 figure
On Polygons Excluding Point Sets
By a polygonization of a finite point set in the plane we understand a
simple polygon having as the set of its vertices. Let and be sets
of blue and red points, respectively, in the plane such that is in
general position, and the convex hull of contains interior blue points
and interior red points. Hurtado et al. found sufficient conditions for the
existence of a blue polygonization that encloses all red points. We consider
the dual question of the existence of a blue polygonization that excludes all
red points . We show that there is a minimal number , which is
polynomial in , such that one can always find a blue polygonization
excluding all red points, whenever . Some other related problems are
also considered.Comment: 14 pages, 15 figure
On k-Convex Polygons
We introduce a notion of -convexity and explore polygons in the plane that
have this property. Polygons which are \mbox{-convex} can be triangulated
with fast yet simple algorithms. However, recognizing them in general is a
3SUM-hard problem. We give a characterization of \mbox{-convex} polygons, a
particularly interesting class, and show how to recognize them in \mbox{} time. A description of their shape is given as well, which leads to
Erd\H{o}s-Szekeres type results regarding subconfigurations of their vertex
sets. Finally, we introduce the concept of generalized geometric permutations,
and show that their number can be exponential in the number of
\mbox{-convex} objects considered.Comment: 23 pages, 19 figure
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