Given a finite set P ⊂ ℝ², the directed Theta-6 graph, denoted Θ₆ (P), is a well-studied geometric graph due to its close relationship with the Delaunay triangulation. The Θ₆ (P) -graph is defined as follows: the plane around each point u ∈ 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 Θ₆ (P) -graph contains an edge from u to v exactly when the interior of ∇ᵤᵛ is disjoint from P, where ∇ᵤᵛ 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 Θ₆ (P) -graph is between 4 and 7 in the worst case (Akitaya, Biniaz, and Bose 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 Θₖ (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.
Bose et al. (2026) studied this question.