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 GP and GQ spanning P and Q, respectively, are called compatible if for every face f in GP , there exists a corresponding face in GQ with the same clockwise ordering of the vertices on its boundary as in f. In particular, GP and GQ must be straightline embeddings of the same connected n-vertex graph G. No polynomial-time algorithm is known for deciding whether two labelled point sets admit compatible geometric graphs. The complexity of the problem is open, even when the graphs are constrained to be triangulations, trees, or simple paths. 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 monotonePeer ReviewedPostprint (published version

On Compatible Matchings

A matching is compatible to two or more labeled point sets of size $n$ with labels $\{1,\dots,n\}$ if its straight-line drawing on each of these point sets is crossing-free. We study the maximum number of edges in a matching compatible to two or more labeled point sets in general position in the plane. We show that for any two labeled convex sets of $n$ points there exists a compatible matching with $\lfloor \sqrt {2n}\rfloor$ edges. More generally, for any $\ell$ labeled point sets we construct compatible matchings of size $\Omega(n^{1/\ell})$. As a corresponding upper bound, we use probabilistic arguments to show that for any $\ell$ given sets of $n$ points there exists a labeling of each set such that the largest compatible matching has ${\mathcal{O}}(n^{2/({\ell}+1)})$ edges. Finally, we show that $\Theta(\log n)$ copies of any set of $n$ points are necessary and sufficient for the existence of a labeling such that any compatible matching consists only of a single edge

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 GP and GQ spanning P and Q, respectively, are called compatible if for every face f in GP , there exists a corresponding face in GQ with the same clockwise ordering of the vertices on its boundary as in f. In particular, GP and GQ must be straightline embeddings of the same connected n-vertex graph G. No polynomial-time algorithm is known for deciding whether two labelled point sets admit compatible geometric graphs. The complexity of the problem is open, even when the graphs are constrained to be triangulations, trees, or simple paths. 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 monotonePeer Reviewe

Compatible Paths on Labelled Point Sets

Let P and Q be finite point sets of the same cardinality in R2, each labelled from 1 to n. Two noncrossing geometric graphs GP and GQ spanning P and Q, respectively, are called compatible if for every face f in GP, there exists a corresponding face in GQ with the same clockwise ordering of the vertices on its boundary as in f. In particular, GP and GQ 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(n2) time for two simple polygons, where the paths are restricted to remain inside the closed polygons; and O(n2logn) time for points in general position if the paths are restricted to be monotone.info:eu-repo/semantics/publishe

Compatible Paths on Labelled Point Sets

Let P and Q be finite point sets of the same cardinality in R2, each labelled from 1 to n. Two noncrossinggeometric graphs GP and GQ spanning P and Q, respectively, are called compatible if for every face f inGP ,there exists a corresponding face in GQ with thesame clockwise ordering of the vertices on its boundaryas in f. In particular, GP and GQ must be straightline embeddings of the same connected n-vertex graphG. No polynomial-time algorithm is known for decidingwhether two labelled point sets admit compatible geometric graphs. The complexity of the problem is open,even when the graphs are constrained to be triangulations, trees, or simple paths.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(n2) time fortwo simple polygons, where the paths are restricted toremain inside the closed polygons; and O(n2log n) timefor points in general position if the paths are restrictedto be monotone.info:eu-repo/semantics/publishe

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 GP and GQ spanning P and Q, respectively, are called compatible if for every face f in GP , there exists a corresponding face in GQ with the same clockwise ordering of the vertices on its boundary as in f. In particular, GP and GQ must be straightline embeddings of the same connected n-vertex graph G. No polynomial-time algorithm is known for deciding whether two labelled point sets admit compatible geometric graphs. The complexity of the problem is open, even when the graphs are constrained to be triangulations, trees, or simple paths. 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 monotonePeer Reviewe