Monotone Paths in Geometric Triangulations

TL;DR

This paper establishes a tighter exponential upper bound on the number of monotone paths in geometric triangulations and provides an efficient algorithm to count such paths in planar geometric graphs.

Contribution

It improves the upper bound from 1.8393^n to 1.7864^n for monotone paths in triangulations and introduces an O(n^2) algorithm for counting monotone paths in planar graphs.

Findings

Maximum monotone paths in triangulations are bounded by O(1.7864^n).

Counting monotone paths in planar graphs can be done in O(n^2) time.

Abstract

(I) We prove that the (maximum) number of monotone paths in a geometric triangulation of $n$ points in the plane is $O(1.7864^n)$. This improves an earlier upper bound of $O(1.8393^n)$; the current best lower bound is $\Omega(1.7003^n)$. (II) Given a planar geometric graph $G$ with $n$ vertices, we show that the number of monotone paths in $G$ can be computed in $O(n^2)$ time.