The Spanning Ratio of the Directed $Θ_6$-Graph is 5

2026-03-09

AI Summary

This work establishes the exact worst-case spanning ratio of the directed $\Theta_6$-graph as 5, marking the first tight bound for any $\Theta_k$-graph. Resolving the long-standing uncertainty within the previously known interval [4,7], the authors achieve a precise characterization by constructing a matching lower-bound instance and proving a complementary upper bound via a novel linear programming approach. Key technical contributions include geometric graph-theoretic analysis, partitioning of equilateral triangular regions, modeling of path structures through convergent series, and an innovative application of linear programming to handle shortest-path problems under angular constraints. This result not only closes a persistent theoretical gap but also introduces a new paradigm for analyzing spanning ratios in $\Theta_k$-graphs.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

Given a finite set $P\subset\mathbb{R}^2$, the directed Theta-6 graph, denoted $\vecΘ_6(P)$, is a well-studied geometric graph due to its close relationship with the Delaunay triangulation. The $\vecΘ_6(P)$-graph is defined as follows: the plane around each point $u\in P$ is partitioned into $6$ equiangular cones with apex $u$, and in each cone, $u$ is joined to the point whose projection on the bisector of the cone is closest. Equivalently, the $\vecΘ_6(P)$-graph contains an edge from $u$ to $v$ exactly when the interior of $\nabla_u^v$ is disjoint from $P$, where $\nabla_u^v$ is the unique equilateral triangle containing $u$ on a corner, $v$ on the opposite side, and whose sides are parallel to the cone boundaries. It was previously shown that the spanning ratio of the $\vecΘ_6(P)$-graph is between $4$ and $7$ in the worst case (Akitaya, Biniaz, and Bose \emph{Comput. Geom.}, 105-106:101881, 2022). We close this gap by showing a tight spanning ratio of 5. This is the first tight bound proven for the spanning ratio of any $\vecΘ_k(P)$-graph. Our lower bound models a long path by mapping it to a converging series. Our upper bound proof uses techniques novel to the area of spanners. We use linear programming to prove that among several candidate paths, there exists a path satisfying our bound.

Problem

• spanning ratio
• Theta-6 graph
• geometric spanner
• Delaunay triangulation
• computational geometry

Innovation

• spanning ratio
• Theta-6 graph
• geometric spanner
• linear programming
• tight bound