  A set of segments in the plane may form a Euclidean TSP tour or a matching,
among others. Optimal TSP tours as well as minimum weight perfect matchings
have no crossing segments, but several heuristics and approximation algorithms
may produce solutions with crossings. To improve such solutions, we can
successively apply a flip operation that replaces a pair of crossing segments
by non-crossing ones. This paper considers the maximum number D(n) of flips
performed on n segments. First, we present reductions relating D(n) for
different sets of segments (TSP tours, monochromatic matchings, red-blue
matchings, and multigraphs). Second, we show that if all except t points are in
convex position, then D(n) = O(tn^2), providing a smooth transition between the
convex O(n^2) bound and the general O(n^3) bound. Last, we show that if instead
of counting the total number of flips, we only count the number of distinct
flips, then the cubic upper bound improves to O(n^{8/3}).
