On spanning properties of various Delaunay graphs

A geometric graph G is a graph whose vertices are points in the plane and whose edges are line segments weighted by the Euclidean distance between their endpoints. In this setting, a t-spanner of G is a connected spanning subgraph G´ with the property that for every pair of vertices x, y, the shortest path from x to y in G´ has weight at most L ≥ 1 times the shortest path from x to y in G. The parameter t is commonly referred to as the spanning ratio or the stretch factor. Among the many beautiful properties that Delaunay graphs possess, a constant spanning ratio is one of them. We provide a comprehensive overview of various results concerning the spanning ratio among other properties of different types of Delaunay graphs and their subgraphs.