Plane Geometric Spanners

Let S be a set of n points in the plane and let G be an undirected graph with vertex set S, in which each edge (u, v) has a weight, which is equal to the Euclidean distance |uv| between the points u and v. For any two points p and q in S, we denote their shortest-path distance in G by δG(p, q). If t ≥ 1 is a real number, then we say that G is a t-spanner for S if δG(p, q) ≤ t|pq| for any two points p and q in S. Thus, if t is close to 1, then the graph G contains close approximations to the ( n 2 ) Euclidean distances determined by the pairs of points in S. If, additionally, G consists of O(n) edges, then this graph can be considered a sparse approximation to the complete graph on S. The smallest value of t for which G is a t-spanner is called the stretch factor (or dilation) of G. For a comprehensive overview of geometric spanners, see the book by Narasimhan and Smid [16]. We assume that each edge (u, v) of G is embedded as the straight-line segment between the points u and v. We say that the graph G is plane if edges only intersect at their common vertices. In this entry, we consider the following two problems: