On path-greedy geometric spanners

A t-spanner is a subgraph of a graph G in which the length of the shortest path between two vertices never exceeds t times the length of the shortest path between them in G. A geometric graph is one whose vertices are points and whose edges are line segments between the corresponding points. Geometric t-spanners are t-spanners of the complete geometric graph on a given point set. Besides approximating the distance between points, we may ask a geometric t-spanner to be planar, have low degree, or low total edge length.