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 γ with 0 < γ < π, we design an O(n)-time algorithm that constructs a connected spanning subgraph G′ of G whose maximum degree is at most 14 + d2π/γe. If G is the Delaunay triangulation of V , and γ = 2π/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 a triangulation satisfying the diamond property, then for a specific range of values of γ 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 γ. If G is the graph consisting of all Delaunay edges of length at most 1, and γ = π/3, we show that a modified version of the algorithm produces a plane subgraph G′ of the unit-disk graph which 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.