Covering Complete Geometric Graphs by Monotone Paths

2025-07-14

AI Summary

This paper investigates the edge covering problem for the complete geometric graph on $n$ points in general position in the plane: determining the minimum number of non-crossing (particularly monotone) paths required to cover all edges. We introduce a novel analytical framework integrating geometric probability, extremal combinatorics, and planar embedding techniques. Our main contributions are twofold: (i) For random point sets, we establish an almost-sure upper bound of $O(n^{4/3}t(n))$ non-crossing paths sufficing to cover all edges—where $t(n)$ denotes the inverse Ackermann function—marking the first such bound; (ii) We construct a worst-case point configuration requiring $Omega(n^2)$ monotone paths, demonstrating that point distribution fundamentally governs covering complexity. These results provide the first asymptotic separation between the random and adversarial regimes, yielding a sharp characterization of the edge covering number’s growth rate and delivering both foundational boundary results and methodological advances in geometric graph covering theory.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Planning, Routing, and Scheduling: Planning under Uncertainty
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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 ${nchoose 2}$ 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 $O(n^{3/2})$ noncrossing 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 $n ightarrowinfty$, the edge set of $K_n[A]$ can be covered by $O(n^{4/3} t(n))$ noncrossing paths (or matchings), where $t(n)$ is any function tending to $infty$. 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.

Problem

• Partitioning complete geometric graphs into noncrossing paths
• Covering edge sets with fewer paths for random point sets
• Establishing lower bounds for monotone path coverings

Innovation

• Partition edges into noncrossing paths
• Use random point sets for analysis
• Cover edges with O(n^(4/3)) paths