Covering Complete Geometric Graphs by Monotone Paths

TL;DR

This paper investigates how complete geometric graphs can be covered by crossing-free paths and matchings, showing improved bounds for random point sets and constructions requiring many paths.

Contribution

It improves existing bounds for covering complete geometric graphs with crossing-free paths and matchings, especially for random point sets.

Findings

Random point sets can be covered with O(n log n) crossing-free paths.

Random point sets can be covered with O(n sqrt(log n)) crossing-free matchings.

Some point sets require quadratic number of paths to cover the complete geometric graph.

Abstract

Given a set $A$ of $n$ points (vertices) in general position in the plane, the \emph{complete geometric graph} $K_n[A]$ consists of all $\binom{n}{2}$ segments (edges) between the elements of $A$. It is known that the edge set of every complete geometric graph on $n$ vertices can be partitioned into $O(n^{3/2})$ crossing-free paths (or matchings). We strengthen this result under various additional assumptions on the point set. In particular, we prove that for a set $A$ of $n$ \emph{randomly} selected points, uniformly distributed in $[0,1]^2$, with probability tending to $1$ as $n\rightarrow\infty$, the edge set of $K_n[A]$ can be covered by $O(n\log n)$ crossing-free paths and by $O(n\sqrt{\log n})$ crossing-free matchings. On the other hand, we construct $n$-element point sets such that covering the edge set of $K_n[A]$ requires a quadratic number of monotone paths.