Compatible Paths on Labelled Point Sets

Let P and Q be finite point sets of the same cardinality in ℝ^2, each labelled from 1 to n. Two noncrossing geometric graphs G_P and G_Q spanning P and Q, respectively, are called compatible if for every face f in G_P, there exists a corresponding face in G_Q with the same clockwise ordering of the vertices on its boundary as in f. In particular, G_P and G_Q must be straight-line embeddings of the same connected n-vertex graph. Deciding whether two labelled point sets admit compatible geometric paths is known to be NP-complete. We give polynomial-time algorithms to find compatible paths or report that none exist in three scenarios: O(n) time for points in convex position; O(n^2) time for two simple polygons, where the paths are restricted to remain inside the closed polygons; and O(n^2 log n) time for points in general position if the paths are restricted to be monotone.