Given a point set $P$ in the Euclidean space, a geometric $t$-spanner $G$ is a graph on $P$ such that for every pair of points, the shortest path in $G$ between those points is at most a factor $t$ longer than the Euclidean distance between those points. The value $t\geq 1$ is called the dilation of $G$. Commonly, the aim is to construct a $t$-spanner with additional desirable properties. In graph theory, a powerful tool to admit efficient algorithms is bounded tree-width. We therefore investigate the problem of computing geometric spanners with bounded tree-width and small dilation $t$. Let $d$ be a fixed integer and $P \subset \mathbb{R}^d$ be a point set with $n$ points. We give a first algorithm to compute an $\mathcal{O}(n/k^{d/(d-1)})$-spanner on $P$ with tree-width at most $k$. The dilation obtained by the algorithm is asymptotically worst-case optimal for graphs with tree-width $k$: We show that there is a set of $n$ points such that every spanner of tree-width $k$ has dilation $\mathcal{O}(n/k^{d/(d-1)})$. We further prove a tight dependency between tree-width and the number of edges in sparse connected planar graphs, which admits, for point sets in $\mathbb{R}^2$, a plane spanner with tree-width at most $k$ and small maximum vertex degree. Finally, we show an almost tight bound on the minimum dilation of a spanning tree of $n$ equally spaced points on a circle, answering an open question asked in previous work.

翻译：给定欧几里得空间中的一个点集$P$，几何$t$-生成图$G$是$P$上的一个图，使得对于任意点对，$G$中这两点间的最短路径长度至多是这两点欧几里得距离的$t$倍。值$t\geq 1$称为$G$的膨胀率。通常目标是构造一个具有额外优良性质的$t$-生成图。在图论中，有界树宽是允许高效算法的一个强大工具。因此，我们研究计算具有有界树宽和小膨胀率$t$的几何生成图的问题。设$d$为固定整数，$P \subset \mathbb{R}^d$为包含$n$个点的点集。我们给出了第一个算法，用于计算$P$上一个树宽至多为$k$的$\mathcal{O}(n/k^{d/(d-1)})$-生成图。该算法获得的膨胀率对于树宽为$k$的图是渐近最坏情况最优的：我们证明存在一个$n$个点的点集，使得每个树宽为$k$的生成图的膨胀率均为$\mathcal{O}(n/k^{d/(d-1)})$。我们进一步证明了稀疏连通平面图中树宽与边数之间的紧密依赖关系，这为$\mathbb{R}^2$中的点集允许存在一个树宽至多为$k$且最大顶点度较小的平面生成图。最后，我们给出了圆上$n$个等间距点的生成树的最小膨胀率的一个几乎紧的界，回答了先前工作中提出的一个开放性问题。