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.

翻译：给定有限点集$P\subset\mathbb{R}^2$，有向Theta-6图（记作$\vecΘ_6(P)$）因其与Delaunay三角剖分的密切关系而成为一个被深入研究的几何图。$\vecΘ_6(P)$图的定义如下：围绕每个点$u\in P$的平面被划分为6个以$u$为顶点的等角锥，在每个锥内，$u$连接到其在锥平分线上投影最近的点。等价地，当且仅当$\nabla_u^v$的内部与$P$不相交时，$\vecΘ_6(P)$图包含一条从$u$到$v$的边，其中$\nabla_u^v$是一个唯一的等边三角形，其一个顶点为$u$，对边上包含$v$，且各边平行于锥边界。先前研究表明，在最坏情况下$\vecΘ_6(P)$图的跨越比介于4到7之间（Akitaya, Biniaz和Bose，《计算几何》，105-106:101881, 2022）。我们通过证明其紧跨越比为5来填补这一空白。这是首次针对任何$\vecΘ_k(P)$图证明的紧界。我们的下界构造通过将长路径映射为收敛级数来实现。上界证明采用了跨度图领域的新技术，我们利用线性规划证明了在若干候选路径中存在满足该界的路径。