Diamond triangulations contain spanners of bounded degree

Given a triangulation G, whose vertex set V is a set of n points in the plane, and given a real number gamma with 0 < gamma < 7, we design an O(n)-time algorithm that constructs a connected spanning subgraph G' of G whose maximum degree is at most 14 + [2 pi/gamma]. If G is the Delaunay triangulation of V, and gamma = 2 pi/3, we show that G' is a t-spanner of V (for some constant t) with maximum degree at most 17, thereby improving the previously best known degree bound of 23. If G is the graph consisting of all Delarmay edges of length at most 1, and gamma = pi/3, we show that G' is a t-spanner (for some constant t) of the unit-disk graph of V, whose maximum degree is at most 20, thereby improving the previously best known degree bound of 25. Finally, if G is a triangulation satisfying the diamond property, then for a specific range of values of gamma dependent on the angle of the diamonds, we show that G' is a t-spanner of V (for some constant t) whose maximum degree is bounded by a constant dependent on gamma.