[Paper Review] Increasing-Chord Graphs On Point Sets

Hooman Reisi Dehkordi, Fabrizio Frati+1 more|arXiv (Cornell University)|Aug 12, 2014

Computational Geometry and Mesh Generation4 citations

TL;DR

This paper presents two main results: first, for any point set P with n points, it constructs a planar increasing-chord graph using O(n) Steiner points; second, for convex point sets, it constructs an increasing-chord graph with O(n log n) edges and no Steiner points. The approach leverages Gabriel triangulations and divide-and-conquer strategies on geometric partitions to ensure increasing-chord paths between all vertex pairs.

ABSTRACT

We tackle the problem of constructing increasing-chord graphs spanning point sets. We prove that, for every point set P with n points, there exists an increasing-chord planar graph with O(n) Steiner points spanning P. Further, we prove that, for every convex point set P with n points, there exists an increasing-chord graph with O(n log n) edges (and with no Steiner points) spanning P.

Motivation & Objective

• To resolve an open problem posed by Alamdari et al. regarding the existence of planar increasing-chord graphs spanning arbitrary point sets using only O(n) Steiner points.
• To construct increasing-chord graphs with sub-quadratic edge counts for convex point sets, improving upon prior constructions that required Steiner points.
• To establish structural properties of Gabriel triangulations that ensure increasing-chord paths, thereby enabling efficient graph construction.
• To explore the feasibility of planar increasing-chord graphs for non-convex point sets and higher-dimensional generalizations.
• To investigate whether increasing-chord graphs with o(n²) edges exist for any point set, particularly non-convex configurations.

Proposed method

• Uses Gabriel triangulations—subgraphs of Delaunay triangulations—as a foundation, leveraging their geometric properties to ensure increasing-chord behavior.
• Applies a recursive divide-and-conquer strategy based on geometric partitions induced by two perpendicular directed lines, ensuring balanced subsets.
• Employs a recursive construction where increasing-chord paths are built between subsets defined by directional partitions, using intermediate Steiner points only when necessary.
• Applies a key lemma to ensure that for any convex point set and a suitable direction, the partition into four quadrants yields balanced subsets, enabling logarithmic recursion depth.
• Combines increasing-chord graphs on complementary subsets (e.g., Q_a ∪ Q_c and Q_b ∪ Q_d) to cover all inter-partition pairs, then recursively constructs missing paths between remaining pairs.
• Uses recurrence relations to bound the number of edges: f(n) ≤ 2n + 2f(n/2 + 1), leading to an O(n log n) edge count for convex point sets.

Experimental results

Research questions

• RQ1 Can a planar increasing-chord graph be constructed for any point set P using only O(n) Steiner points?
• RQ2 Does every convex point set admit an increasing-chord graph with O(n log n) edges and no Steiner points?
• RQ3 Are Gabriel triangulations in R² always increasing-chord graphs, and can this property be generalized to higher dimensions?
• RQ4 Is it possible to construct increasing-chord graphs with o(n²) edges for arbitrary (non-convex) point sets?
• RQ5 Can planar increasing-chord graphs be constructed for point sets lying on three or more lines, particularly on the boundary of an acute triangle?

Key findings

• For any point set P of size n, there exists a planar increasing-chord graph with O(n) Steiner points spanning P, answering an open question from Alamdari et al.
• For every convex point set P of size n, there exists an increasing-chord graph with O(n log n) edges and no Steiner points, constructed via recursive geometric partitioning.
• Gabriel triangulations of point sets in R² are shown to be increasing-chord graphs, providing a new class of proximity graphs with strong routing properties.
• The recursive construction achieves a recurrence f(n) ≤ 2n + 2f(n/2 + 1), which solves to f(n) ∈ O(n log n), proving the edge bound for convex point sets.
• The method ensures that every pair of vertices (including Steiner points) has an increasing-chord path in both directions, satisfying the full increasing-chord condition.
• The construction is effective even for non-convex sets when Steiner points are allowed, but the existence of planar increasing-chord graphs without Steiner points for general point sets remains open.