47,703 research outputs found
Drawing the double circle on a grid of minimum size
In 1926, Jarník introduced the problem of drawing a convex n-gon with vertices having integer coordinates. He constructed such a drawing in the grid [1, c ·n 3/2]2 for some constant c > 0, and showed that this grid
size is optimal up to a constant factor. We consider the analogous problem of drawing the double circle, and prove that it can be done within the same grid size. Moreover, we give an O(n log n)-time algorithm to construct such a point set.Consejo Nacional de Ciencia y Tecnologia (México)Programa de Apoyo a Proyectos de Investigación e Innovación Tecnológica (Universidad Nacional Autónoma de México)Comisión Nacional de Investigación Científica y Tecnológica (Chile)Fondo Nacional de Desarrollo Científico y Tecnológico (Chile
Grid-Obstacle Representations with Connections to Staircase Guarding
In this paper, we study grid-obstacle representations of graphs where we
assign grid-points to vertices and define obstacles such that an edge exists if
and only if an -monotone grid path connects the two endpoints without
hitting an obstacle or another vertex. It was previously argued that all planar
graphs have a grid-obstacle representation in 2D, and all graphs have a
grid-obstacle representation in 3D. In this paper, we show that such
constructions are possible with significantly smaller grid-size than previously
achieved. Then we study the variant where vertices are not blocking, and show
that then grid-obstacle representations exist for bipartite graphs. The latter
has applications in so-called staircase guarding of orthogonal polygons; using
our grid-obstacle representations, we show that staircase guarding is
\textsc{NP}-hard in 2D.Comment: To appear in the proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Drawing the Horton Set in an Integer Grid of Minimum Size
In 1978 Erd\H os asked if every sufficiently large set of points in general
position in the plane contains the vertices of a convex -gon, with the
additional property that no other point of the set lies in its interior.
Shortly after, Horton provided a construction---which is now called the Horton
set---with no such -gon. In this paper we show that the Horton set of
points can be realized with integer coordinates of absolute value at most
. We also show that any set of points
with integer coordinates combinatorially equivalent (with the same order type)
to the Horton set, contains a point with a coordinate of absolute value at
least , where is a positive constant
Flip Distance Between Triangulations of a Simple Polygon is NP-Complete
Let T be a triangulation of a simple polygon. A flip in T is the operation of
removing one diagonal of T and adding a different one such that the resulting
graph is again a triangulation. The flip distance between two triangulations is
the smallest number of flips required to transform one triangulation into the
other. For the special case of convex polygons, the problem of determining the
shortest flip distance between two triangulations is equivalent to determining
the rotation distance between two binary trees, a central problem which is
still open after over 25 years of intensive study. We show that computing the
flip distance between two triangulations of a simple polygon is NP-complete.
This complements a recent result that shows APX-hardness of determining the
flip distance between two triangulations of a planar point set.Comment: Accepted versio
Graphic arts techniques and equipment - A compilation
Summary descriptions of NASA graphic arts techniques and equipmen
On Upward Drawings of Trees on a Given Grid
Computing a minimum-area planar straight-line drawing of a graph is known to
be NP-hard for planar graphs, even when restricted to outerplanar graphs.
However, the complexity question is open for trees. Only a few hardness results
are known for straight-line drawings of trees under various restrictions such
as edge length or slope constraints. On the other hand, there exist
polynomial-time algorithms for computing minimum-width (resp., minimum-height)
upward drawings of trees, where the height (resp., width) is unbounded.
In this paper we take a major step in understanding the complexity of the
area minimization problem for strictly-upward drawings of trees, which is one
of the most common styles for drawing rooted trees. We prove that given a
rooted tree and a grid, it is NP-hard to decide whether
admits a strictly-upward (unordered) drawing in the given grid.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
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