3,566 research outputs found
On Universal Point Sets for Planar Graphs
A set P of points in R^2 is n-universal, if every planar graph on n vertices
admits a plane straight-line embedding on P. Answering a question by Kobourov,
we show that there is no n-universal point set of size n, for any n>=15.
Conversely, we use a computer program to show that there exist universal point
sets for all n<=10 and to enumerate all corresponding order types. Finally, we
describe a collection G of 7'393 planar graphs on 35 vertices that do not admit
a simultaneous geometric embedding without mapping, that is, no set of 35
points in the plane supports a plane straight-line embedding of all graphs in
G.Comment: Fixed incorrect numbers of universal point sets in the last par
Recognizing and Drawing IC-planar Graphs
IC-planar graphs are those graphs that admit a drawing where no two crossed
edges share an end-vertex and each edge is crossed at most once. They are a
proper subfamily of the 1-planar graphs. Given an embedded IC-planar graph
with vertices, we present an -time algorithm that computes a
straight-line drawing of in quadratic area, and an -time algorithm
that computes a straight-line drawing of with right-angle crossings in
exponential area. Both these area requirements are worst-case optimal. We also
show that it is NP-complete to test IC-planarity both in the general case and
in the case in which a rotation system is fixed for the input graph.
Furthermore, we describe a polynomial-time algorithm to test whether a set of
matching edges can be added to a triangulated planar graph such that the
resulting graph is IC-planar
Fast and Compact Exact Distance Oracle for Planar Graphs
For a given a graph, a distance oracle is a data structure that answers
distance queries between pairs of vertices. We introduce an -space
distance oracle which answers exact distance queries in time for
-vertex planar edge-weighted digraphs. All previous distance oracles for
planar graphs with truly subquadratic space i.e., space
for some constant ) either required query time polynomial in
or could only answer approximate distance queries.
Furthermore, we show how to trade-off time and space: for any , we show how to obtain an -space distance oracle that answers
queries in time . This is a polynomial
improvement over the previous planar distance oracles with query
time
Cubic Augmentation of Planar Graphs
In this paper we study the problem of augmenting a planar graph such that it
becomes 3-regular and remains planar. We show that it is NP-hard to decide
whether such an augmentation exists. On the other hand, we give an efficient
algorithm for the variant of the problem where the input graph has a fixed
planar (topological) embedding that has to be preserved by the augmentation. We
further generalize this algorithm to test efficiently whether a 3-regular
planar augmentation exists that additionally makes the input graph connected or
biconnected. If the input graph should become even triconnected, we show that
the existence of a 3-regular planar augmentation is again NP-hard to decide.Comment: accepted at ISAAC 201
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