5,347 research outputs found
Compatible matchings in geometric graphs
Two non-crossing geometric graphs on the same set of points are compatible if their union
is also non-crossing. In this paper, we prove that every graph G that has an outerplanar embedding
admits a non-crossing perfect matching compatible with G. Moreover, for non-crossing geometric trees
and simple polygons, we study bounds on the minimum number of edges that a compatible non-crossing
perfect matching must share with the tree or the polygon. We also give bounds on the maximal size of
a compatible matching (not necessarily perfect) that is disjoint from the tree or the polygon.Postprint (published version
Counting and Enumerating Crossing-free Geometric Graphs
We describe a framework for counting and enumerating various types of
crossing-free geometric graphs on a planar point set. The framework generalizes
ideas of Alvarez and Seidel, who used them to count triangulations in time
where is the number of points. The main idea is to reduce the
problem of counting geometric graphs to counting source-sink paths in a
directed acyclic graph.
The following new results will emerge. The number of all crossing-free
geometric graphs can be computed in time for some .
The number of crossing-free convex partitions can be computed in time
. The number of crossing-free perfect matchings can be computed in
time . The number of convex subdivisions can be computed in time
. The number of crossing-free spanning trees can be computed in time
for some . The number of crossing-free spanning cycles
can be computed in time for some .
With the same bounds on the running time we can construct data structures
which allow fast enumeration of the respective classes. For example, after
time of preprocessing we can enumerate the set of all crossing-free
perfect matchings using polynomial time per enumerated object. For
crossing-free perfect matchings and convex partitions we further obtain
enumeration algorithms where the time delay for each (in particular, the first)
output is bounded by a polynomial in .
All described algorithms are comparatively simple, both in terms of their
analysis and implementation
Characterization of co-blockers for simple perfect matchings in a convex geometric graph
Consider the complete convex geometric graph on vertices, ,
i.e., the set of all boundary edges and diagonals of a planar convex -gon
. In [C. Keller and M. Perles, On the Smallest Sets Blocking Simple Perfect
Matchings in a Convex Geometric Graph], the smallest sets of edges that meet
all the simple perfect matchings (SPMs) in (called "blockers") are
characterized, and it is shown that all these sets are caterpillar graphs with
a special structure, and that their total number is . In this
paper we characterize the co-blockers for SPMs in , that is, the
smallest sets of edges that meet all the blockers. We show that the co-blockers
are exactly those perfect matchings in where all edges are of odd
order, and two edges of that emanate from two adjacent vertices of
never cross. In particular, while the number of SPMs and the number of blockers
grow exponentially with , the number of co-blockers grows
super-exponentially.Comment: 8 pages, 4 figure
Maximum Matchings in Geometric Intersection Graphs
Let G be an intersection graph of n geometric objects in the plane. We show that a maximum matching in G can be found in O(Ď3Ď/2nĎ/2) time with high probability, where Ď is the density of the geometric objects and Ď>2 is a constant such that nĂn matrices can be multiplied in O(nĎ) time. The same result holds for any subgraph of G, as long as a geometric representation is at hand. For this, we combine algebraic methods, namely computing the rank of a matrix via Gaussian elimination, with the fact that geometric intersection graphs have small separators. We also show that in many interesting cases, the maximum matching problem in a general geometric intersection graph can be reduced to the case of bounded density. In particular, a maximum matching in the intersection graph of any family of translates of a convex object in the plane can be found in O(nĎ/2) time with high probability, and a maximum matching in the intersection graph of a family of planar disks with radii in [1,Ψ] can be found in O(Ψ6log11n+Ψ12ĎnĎ/2) time with high probability
Packing Plane Perfect Matchings into a Point Set
Given a set of points in the plane, where is even, we consider
the following question: How many plane perfect matchings can be packed into
? We prove that at least plane perfect matchings
can be packed into any point set . For some special configurations of point
sets, we give the exact answer. We also consider some extensions of this
problem
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
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