105 research outputs found
Geodesic-Preserving Polygon Simplification
Polygons are a paramount data structure in computational geometry. While the
complexity of many algorithms on simple polygons or polygons with holes depends
on the size of the input polygon, the intrinsic complexity of the problems
these algorithms solve is often related to the reflex vertices of the polygon.
In this paper, we give an easy-to-describe linear-time method to replace an
input polygon by a polygon such that (1)
contains , (2) has its reflex
vertices at the same positions as , and (3) the number of vertices
of is linear in the number of reflex vertices. Since the
solutions of numerous problems on polygons (including shortest paths, geodesic
hulls, separating point sets, and Voronoi diagrams) are equivalent for both
and , our algorithm can be used as a preprocessing
step for several algorithms and makes their running time dependent on the
number of reflex vertices rather than on the size of
Linear transformation distance for bichromatic matchings
Let be a set of points in general position, where is a
set of blue points and a set of red points. A \emph{-matching}
is a plane geometric perfect matching on such that each edge has one red
endpoint and one blue endpoint. Two -matchings are compatible if their
union is also plane.
The \emph{transformation graph of -matchings} contains one node for each
-matching and an edge joining two such nodes if and only if the
corresponding two -matchings are compatible. In SoCG 2013 it has been shown
by Aloupis, Barba, Langerman, and Souvaine that this transformation graph is
always connected, but its diameter remained an open question. In this paper we
provide an alternative proof for the connectivity of the transformation graph
and prove an upper bound of for its diameter, which is asymptotically
tight
Different Types of Isomorphisms of Drawings of Complete Multipartite Graphs
Simple drawings are drawings of graphs in which any two edges intersect at
most once (either at a common endpoint or a proper crossing), and no edge
intersects itself. We analyze several characteristics of simple drawings of
complete multipartite graphs: which pairs of edges cross, in which order they
cross, and the cyclic order around vertices and crossings, respectively. We
consider all possible combinations of how two drawings can share some
characteristics and determine which other characteristics they imply and which
they do not imply. Our main results are that for simple drawings of complete
multipartite graphs, the orders in which edges cross determine all other
considered characteristics. Further, if all partition classes have at least
three vertices, then the pairs of edges that cross determine the rotation
system and the rotation around the crossings determine the extended rotation
system. We also show that most other implications -- including the ones that
hold for complete graphs -- do not hold for complete multipartite graphs. Using
this analysis, we establish which types of isomorphisms are meaningful for
simple drawings of complete multipartite graphs.Comment: Appears in the Proceedings of the 31st International Symposium on
Graph Drawing and Network Visualization (GD 2023
Adjacency Graphs of Polyhedral Surfaces
We study whether a given graph can be realized as an adjacency graph of the
polygonal cells of a polyhedral surface in . We show that every
graph is realizable as a polyhedral surface with arbitrary polygonal cells, and
that this is not true if we require the cells to be convex. In particular, if
the given graph contains , , or any nonplanar -tree as a
subgraph, no such realization exists. On the other hand, all planar graphs,
, and can be realized with convex cells. The same holds for
any subdivision of any graph where each edge is subdivided at least once, and,
by a result from McMullen et al. (1983), for any hypercube.
Our results have implications on the maximum density of graphs describing
polyhedral surfaces with convex cells: The realizability of hypercubes shows
that the maximum number of edges over all realizable -vertex graphs is in
. From the non-realizability of , we obtain that
any realizable -vertex graph has edges. As such, these graphs
can be considerably denser than planar graphs, but not arbitrarily dense.Comment: To appear in Proc. SoCG 202
Shooting Stars in Simple Drawings of
Simple drawings are drawings of graphs in which two edges have at most one
common point (either a common endpoint, or a proper crossing). It has been an
open question whether every simple drawing of a complete bipartite graph
contains a plane spanning tree as a subdrawing. We answer this
question to the positive by showing that for every simple drawing of
and for every vertex in that drawing, the drawing contains a shooting star
rooted at , that is, a plane spanning tree containing all edges incident to
.Comment: Appears in the Proceedings of the 30th International Symposium on
Graph Drawing and Network Visualization (GD 2022
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