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Covering Complete Geometric Graphs by Monotone Paths

Adrian Dumitrescu, János Pach, Morteza Saghafian, Alex Scott

TL;DR

The paper investigates decomposing the edge set of the complete geometric graph $K_n[A]$ on $n$ points into crossing-free monotone paths and monotone matchings. It proves that for random point sets in $[0,1]^2$, with high probability the edges can be covered by $O(n ext{log} n)$ monotone paths (hence crossing-free) and by $O(n ext{sqrt} { ext{log}} n)$ monotone matchings, with an even stronger bound of $O(n ext{log} n)$ monotone paths. For $ extalpha$-dense point sets, the edge set is coverable by $O(n^{3/2})$ monotone paths and by $O(n^{3/2})$ monotone matchings, while there exist worst-case point sets requiring $ heta(n^2)$ monotone paths. The paper also provides lower and upper bounds for arbitrary point sets, including a quadratic lower bound via a tripartite construction and a near-quadratic upper bound via a packing argument using $K_6$ subgraphs and zig-zag paths. These results advance understanding of geometric graph decompositions, with implications for monotone structure in typical versus adversarial point configurations and for related extremal questions.

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.

Covering Complete Geometric Graphs by Monotone Paths

TL;DR

The paper investigates decomposing the edge set of the complete geometric graph on points into crossing-free monotone paths and monotone matchings. It proves that for random point sets in , with high probability the edges can be covered by monotone paths (hence crossing-free) and by monotone matchings, with an even stronger bound of monotone paths. For -dense point sets, the edge set is coverable by monotone paths and by monotone matchings, while there exist worst-case point sets requiring monotone paths. The paper also provides lower and upper bounds for arbitrary point sets, including a quadratic lower bound via a tripartite construction and a near-quadratic upper bound via a packing argument using subgraphs and zig-zag paths. These results advance understanding of geometric graph decompositions, with implications for monotone structure in typical versus adversarial point configurations and for related extremal questions.

Abstract

Given a set of points (vertices) in general position in the plane, the \emph{complete geometric graph} consists of all segments (edges) between the elements of . It is known that the edge set of every complete geometric graph on vertices can be partitioned into 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 of \emph{randomly} selected points, uniformly distributed in , with probability tending to as , the edge set of can be covered by crossing-free paths and by crossing-free matchings. On the other hand, we construct -element point sets such that covering the edge set of requires a quadratic number of monotone paths.

Paper Structure

This paper contains 5 sections, 5 theorems, 6 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm:random']}
  3. Proof of Theorem \ref{['thm:dense']}
  4. Proof of Theorem \ref{['thm:arbitrary']}
  5. Concluding remarks

Key Result

Theorem 2

Let $A$ be a set of $n$ random points uniformly distributed in $[0,1]^2$, and let $n\rightarrow\infty$. Then, with probability tending to $1$, the edge set of $K_n[A]$ can be covered by at most $O(n\log n)$ crossing-free paths, and by $O(n\sqrt{\log n})$ crossing-free matchings.

Figures (3)

  • Figure 1: Left: four lines making an angle of $\theta$ with $ab$ through $a$ and $b$. Center: the geometric locus (shaded) associated with a vertical segment. Right: a bounding rectangle for the locus associated with a long vertical segment.
  • Figure 2: Covering the edge set of a tripartite graph by monotone paths. The figure shows the edges in $E_0$ corresponding to a monotone path $\xi$; note that this edge set may be disconnected. A direction of monotonicity is drawn in blue color.
  • Figure 3: Covering the edge set of the complete graph on six points by $3$-edge zig-zag paths.

Theorems & Definitions (7)

  • Theorem 2
  • Definition 3
  • Theorem 4
  • Conjecture 5
  • Theorem 6
  • Lemma 7
  • Lemma 8: Wilson 1975