Counting Plane Graphs: Flippability and its Applications

TL;DR

This paper introduces pseudo-simultaneously flippable edges in triangulations, establishes bounds on their quantity, and derives new exponential upper bounds for the number of crossing-free graphs, spanning trees, and forests on planar point sets.

Contribution

It generalizes flippability concepts, proves tight bounds on pseudo-simultaneously flippable edges, and applies these to improve upper bounds on crossing-free graphs.

Findings

Established tight lower bounds for pseudo-simultaneously flippable edges.

Derived new exponential upper bounds for crossing-free graphs, spanning trees, and forests.

Provided bounds for graphs with various edge count constraints.

Abstract

We generalize the notions of flippable and simultaneously flippable edges in a triangulation of a set S of points in the plane to so-called \emph{pseudo-simultaneously flippable edges}. Such edges are related to the notion of convex decompositions spanned by S. We prove a worst-case tight lower bound for the number of pseudo-simultaneously flippable edges in a triangulation in terms of the number of vertices. We use this bound for deriving new upper bounds for the maximal number of crossing-free straight-edge graphs that can be embedded on any fixed set of N points in the plane. We obtain new upper bounds for the number of spanning trees and forests as well. Specifically, let tr(N) denote the maximum number of triangulations on a set of N points in the plane. Then we show (using the known bound tr(N) < 30^N) that any N-element point set admits at most 6.9283^N * tr(N) < 207.85^N crossing-free straight-edge graphs, O(4.7022^N) * tr(N) = O(141.07^N) spanning trees, and O(5.3514^N) * tr(N) = O(160.55^N) forests. We also obtain upper bounds for the number of crossing-free straight-edge graphs that have cN, fewer than cN, or more than cN edges, for any constant parameter c, in terms of c and N.