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Partitioning Complete Geometric Graphs on Dense Point Sets into Plane Subgraphs

Adrian Dumitrescu, János Pach

TL;DR

This work addresses the Bose–Hurtado–Rivera-Campo–Wood question for α-dense point sets by proving that the edge-set of the complete geometric graph on such a set can be partitioned into a strictly sublinear fraction of linear-size plane subgraphs. The authors reduce the problem to decomposing a large subset $B$ into a small number of plane subgraphs by exploiting a four-block configuration $B=B_1\cup B_2\cup B_3\cup B_4$ with the inside-triangle property, enabling a decomposition of $K_{4m}[B]$ into $3m$ plane star-forests. They then show α-dense sets contain such a $B$ of linear size and derive an explicit bound $c(\alpha)=1-1/(3k^2(\alpha))$, with $k(\alpha)$ chosen as a function of density. The paper further discusses corollaries for random point sets and situates the results within broader themes on crossing families, halving lines, and geometric thickness, highlighting both improvements and remaining open questions on the density dependence of the bound. This advances understanding of near-linear decompositions in dense geometric graphs and provides a constructive path toward near-optimal plane-partitions in this regime.

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 $Θ(\sqrt{n})$.

Partitioning Complete Geometric Graphs on Dense Point Sets into Plane Subgraphs

TL;DR

This work addresses the Bose–Hurtado–Rivera-Campo–Wood question for α-dense point sets by proving that the edge-set of the complete geometric graph on such a set can be partitioned into a strictly sublinear fraction of linear-size plane subgraphs. The authors reduce the problem to decomposing a large subset into a small number of plane subgraphs by exploiting a four-block configuration with the inside-triangle property, enabling a decomposition of into plane star-forests. They then show α-dense sets contain such a of linear size and derive an explicit bound , with chosen as a function of density. The paper further discusses corollaries for random point sets and situates the results within broader themes on crossing families, halving lines, and geometric thickness, highlighting both improvements and remaining open questions on the density dependence of the bound. This advances understanding of near-linear decompositions in dense geometric graphs and provides a constructive path toward near-optimal plane-partitions in this regime.

Abstract

A \emph{complete geometric graph} consists of a set of 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 , such that every complete geometric graph on points can be partitioned into at most plane graphs (that is, noncrossing subgraphs). We answer this question in the affirmative in the special case where the underlying point set is \emph{dense}, which means that the ratio between the maximum and the minimum distances in is of the order of .
Paper Structure (6 sections, 10 theorems, 10 equations, 4 figures)

This paper contains 6 sections, 10 theorems, 10 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm:partition']}
  3. Proof of Lemma \ref{['lem:4rich']}.
  4. Final argument.
  5. Note.
  6. Concluding remarks

Key Result

Theorem 2

Let $A$ be an $\alpha$-dense point set of $n$ points in general position in the plane, and let $K_n=K_n[A]$ denote the complete geometric graph induced by $A$. Then (the edge set of) $K_n$ can be decomposed into at most $cn$ plane subgraphs, where $c=c(\alpha)<1$ is a constant. Specifically, we have Each of these plane graphs is either a star or a plane union of two stars.

Figures (4)

  • Figure 2: Left: A $5 \times 5$ subgrid of rich cells (shaded) and a relevant subset of four cells. Right: A $5 \times 5$ subgrid of rich cells with some separation.
  • Figure 3: Left: The set of rich cells in $Q$ (each rich cell is shaded). Center: the star triangulation $K$ from a boundary cell in $\mathcal{C}$. Here $|\mathcal{R}|=22$ and $|\mathcal{C}|=7$. Segments in $\mathcal{S}$ are in bold lines. Right: a set of four rich cells as in Lemma \ref{['lem:4rich']}.
  • Figure 4: A dense set of $24$ points with a crossing family of size $12$. The origin (marked with a cross) is not part of the set.
  • Figure 5: The four distinguished subsquares are shaded.

Theorems & Definitions (11)

  • Theorem 2
  • Corollary 3
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • Lemma 10
  • Theorem 11
  • Claim 12
  • ...and 1 more