Graphs of non-crossing perfect matchings

Let Pn be a set of n = 2m points that are the vertices of a convex polygon, and let Mm be the graph having as vertices all the perfect matchings in the point set Pn whose edges are straight line segments and do not cross, and edges joining two perfect matchings M1 and M2 if M2 = M1 ¡ (a; b) ¡ (c; d) + (a; d) + (b; c) for some points a; b; c; d of Pn. We prove the following results about Mm: its diameter is m ¡ 1; it is bipartite for every m; the connectivity is equal to m ¡ 1; it has no Hamilton path for m odd, m > 3; and finally it has a Hamilton cycle for every m even, m>=4

Bounds on the maximum multiplicity of some common geometric graphs

We obtain new lower and upper bounds for the maximum multiplicity of some weighted and, respectively, non-weighted common geometric graphs drawn on n points in the plane in general position (with no three points collinear): perfect matchings, spanning trees, spanning cycles (tours), and triangulations. (i) We present a new lower bound construction for the maximum number of triangulations a set of n points in general position can have. In particular, we show that a generalized double chain formed by two almost convex chains admits {\Omega}(8.65^n) different triangulations. This improves the bound {\Omega}(8.48^n) achieved by the double zig-zag chain configuration studied by Aichholzer et al. (ii) We present a new lower bound of {\Omega}(12.00^n) for the number of non-crossing spanning trees of the double chain composed of two convex chains. The previous bound, {\Omega}(10.42^n), stood unchanged for more than 10 years. (iii) Using a recent upper bound of 30^n for the number of triangulations, due to Sharir and Sheffer, we show that n points in the plane in general position admit at most O(68.62^n) non-crossing spanning cycles. (iv) We derive lower bounds for the number of maximum and minimum weighted geometric graphs (matchings, spanning trees, and tours). We show that the number of shortest non-crossing tours can be exponential in n. Likewise, we show that both the number of longest non-crossing tours and the number of longest non-crossing perfect matchings can be exponential in n. Moreover, we show that there are sets of n points in convex position with an exponential number of longest non-crossing spanning trees. For points in convex position we obtain tight bounds for the number of longest and shortest tours. We give a combinatorial characterization of the longest tours, which leads to an O(nlog n) time algorithm for computing them

Peeling and Nibbling the Cactus: Subexponential-Time Algorithms for Counting Triangulations and Related Problems

Given a set of n points S in the plane, a triangulation T of S is a maximal set of non-crossing segments with endpoints in S. We present an algorithm that computes the number of triangulations on a given set of n points in time n^{ (11+ o(1)) sqrt{n} }, significantly improving the previous best running time of O(2^n n^2) by Alvarez and Seidel [SoCG 2013]. Our main tool is identifying separators of size O(sqrt{n}) of a triangulation in a canonical way. The definition of the separators are based on the decomposition of the triangulation into nested layers ("cactus graphs"). Based on the above algorithm, we develop a simple and formal framework to count other non-crossing straight-line graphs in n^{O(sqrt{n})} time. We demonstrate the usefulness of the framework by applying it to counting non-crossing Hamilton cycles, spanning trees, perfect matchings, 3-colorable triangulations, connected graphs, cycle decompositions, quadrangulations, 3-regular graphs, and more

Peeling and nibbling the cactus: Subexponential-time algorithms for counting triangulations and related problems

Given a set of $n$ points $S$ in the plane, a triangulation $T$ of $S$ is a maximal set of non-crossing segments with endpoints in $S$. We present an algorithm that computes the number of triangulations on a given set of $n$ points in time $n^{(11+ o(1))\sqrt{n} }$, significantly improving the previous best running time of $O(2^n n^2)$ by Alvarez and Seidel [SoCG 2013]. Our main tool is identifying separators of size $O(\sqrt{n})$ of a triangulation in a canonical way. The definition of the separators are based on the decomposition of the triangulation into nested layers ("cactus graphs"). Based on the above algorithm, we develop a simple and formal framework to count other non-crossing straight-line graphs in $n^{O(\sqrt{n})}$ time. We demonstrate the usefulness of the framework by applying it to counting non-crossing Hamilton cycles, spanning trees, perfect matchings, $3$-colorable triangulations, connected graphs, cycle decompositions, quadrangulations, $3$-regular graphs, and more.Comment: 47 pages, 23 Figures, to appear in SoCG 201

Edge-Removal and Non-Crossing Configurations in Geometric Graphs

A geometric graph is a graph G = (V;E) drawn in the plane, such that V is a point set in general position and E is a set of straight-line segments whose endpoints belong to V . We study the following extremal problem for geometric graphs: How many arbitrary edges can be removed from a complete geometric graph with n vertices such that the remaining graph still contains a certain non-crossing subgraph. The non-crossing subgraphs that we consider are perfect matchings, subtrees of a given size, and triangulations. In each case, we obtain tight bounds on the maximum number of removable edges.Postprint (published version

On flips in planar matchings

In this paper we investigate the structure of flip graphs on non-crossing perfect matchings in the plane. Consider all non-crossing straight-line perfect matchings on a set of $2n$ points that are placed equidistantly on the unit circle. The graph~$\mathcal{H}_n$ has those matchings as vertices, and an edge between any two matchings that differ in replacing two matching edges that span an empty quadrilateral with the other two edges of the quadrilateral, provided that the quadrilateral contains the center of the unit circle. We show that the graph~$\mathcal{H}_n$ is connected for odd~$n$, but has exponentially many small connected components for even~$n$, which we characterize and count via Catalan and generalized Narayana numbers. For odd $n$, we also prove that the diameter of~$\mathcal{H}_n$ is linear in~$n$. Furthermore, we determine the minimum and maximum degree of~$\mathcal{H}_n$ for all~$n$, and characterize and count the corresponding vertices. Our results imply the non-existence of certain rainbow cycles, and they answer several open questions and conjectures raised in a recent paper by Felsner, Kleist, M\"utze, and Sering

Packing Plane Perfect Matchings into a Point Set

Given a set $P$ of $n$ points in the plane, where $n$ is even, we consider the following question: How many plane perfect matchings can be packed into $P$? We prove that at least $\lceil\log_2{n}\rceil-2$ plane perfect matchings can be packed into any point set $P$. For some special configurations of point sets, we give the exact answer. We also consider some extensions of this problem