On determining non-isotopic configurations of points on a circle

Given a set P of 2n colored points on a circle O, a configuration of P is a set B of n non-intersecting chords of O such that each chord passes through two points in P of the same color. Two configurations B₁ and B₂ of P are isotopic if we can move, enlarge, or shrink the chords in B₁ (no two chords may contact each other during the process) so that the resulting configuration is identical to B₂. We describe linear time algorithms for determining if P has a configuration and if P has at least two non-isotopic configurations.