Partitioning Complete Geometric Graphs on Dense Point Sets into Plane Subgraphs

TL;DR

This paper proves that dense point sets in the plane allow partitioning their complete geometric graphs into a linear number of noncrossing plane subgraphs, advancing understanding of geometric graph decompositions.

Contribution

It establishes that for dense point sets, the complete geometric graph can be partitioned into at most a constant times n plane subgraphs, confirming a special case of a longstanding open problem.

Findings

Partitioning into at most c*n plane graphs for dense sets

Affirmative answer to a special case of a known question

Density condition is key for the partitioning result

Abstract

A \emph{complete geometric graph} consists of a set $P$ of $n$ points in the plane, in general position, and all segments (edges) connecting them. It is a well known question of Bose, Hurtado, Rivera-Campo, and Wood, whether there exists a positive constant $c<1$, such that every complete geometric graph on $n$ points can be partitioned into at most $cn$ plane graphs (that is, noncrossing subgraphs). We answer this question in the affirmative in the special case where the underlying point set $P$ is \emph{dense}, which means that the ratio between the maximum and the minimum distances in $P$ is of the order of $\Theta(\sqrt{n})$.