Summary: Long non-crossing configurations in the plane (Draft)
It is shown that for any set of 2n points in general position in the plane there is a non-crossing
perfect matching of n straight line segments whose total length is at least 2/ of the maximum
possible total length of a (possibly crossing) perfect matching on these points. The constant
2/ is best possible and a non-crossing matching whose length is at least as above can be found
in polynomial time. Similar results are obtained for the problem of finding a long non-crossing
Hamilton path and a long non-crossing spanning tree for a given set of points in the plane.
A geometric graph is a pair G = (V, E) , where V is a finite set of points in general position in the
plane and E is a family of closed straight line-segments whose end-points lie in V . The elements of
V are called vertices and these of E are called edges. The length of G, denoted by L(G), is the sum
of Euclidean lengths of all edges of G. G is non-crossing if the interiors of all its edges are pairwise
Several results in Combinatorial and Computational Geometry deal with the extremal values of
the length L(G) of a geometric graph G of a prescribed type on a given set of vertices in the plane.
The best known example of a problem of this type is the (Euclidean) travelling salesman problem