7,997 research outputs found
Edge-Removal and Non-Crossing Configurations in Geometric Graphs
A geometric graph is a graph G = (V;E) drawn in the plane, such that V is a point set in general position and E is
a set of straight-line segments whose endpoints belong to V . We study the following extremal problem for geometric
graphs: How many arbitrary edges can be removed from a complete geometric graph with n vertices such that the
remaining graph still contains a certain non-crossing subgraph. The non-crossing subgraphs that we consider are
perfect matchings, subtrees of a given size, and triangulations. In each case, we obtain tight bounds on the maximum
number of removable edges.Postprint (published version
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
The polytope of non-crossing graphs on a planar point set
For any finite set \A of points in , we define a
-dimensional simple polyhedron whose face poset is isomorphic to the
poset of ``non-crossing marked graphs'' with vertex set \A, where a marked
graph is defined as a geometric graph together with a subset of its vertices.
The poset of non-crossing graphs on \A appears as the complement of the star
of a face in that polyhedron.
The polyhedron has a unique maximal bounded face, of dimension
where is the number of points of \A in the interior of \conv(\A). The
vertices of this polytope are all the pseudo-triangulations of \A, and the
edges are flips of two types: the traditional diagonal flips (in
pseudo-triangulations) and the removal or insertion of a single edge.
As a by-product of our construction we prove that all pseudo-triangulations
are infinitesimally rigid graphs.Comment: 28 pages, 16 figures. Main change from v1 and v2: Introduction has
been reshape
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