The Spanning Ratio of the Directed $\Theta_6$-Graph is 5

TL;DR

This paper proves that the directed Theta-6 graph has a tight spanning ratio of 5, closing a previously known gap between 4 and 7, and introduces novel proof techniques for this geometric graph property.

Contribution

The paper establishes the first tight bound of 5 for the spanning ratio of the directed Theta-6 graph, advancing understanding of geometric spanners.

Findings

Spanning ratio of the directed Theta-6 graph is exactly 5.

Introduces novel techniques using linear programming for bounding paths.

Provides the first tight bound for any directed Theta_k-graph.

Abstract

Given a finite set $P\subset\mathbb{R}^2$, the directed Theta-6 graph, denoted $\vec{\Theta}_6(P)$, is a well-studied geometric graph due to its close relationship with the Delaunay triangulation. The $\vec{\Theta}_6(P)$-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 $\vec{\Theta}_6(P)$-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 $\vec{\Theta}_6(P)$-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 $\vec{\Theta}_k(P)$-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.