We present improved upper bounds on the spanning ratio of constrained -graphs with at least 6 cones and constrained Yao-graphs with 5 or at least 7 cones. Given a set of points in the plane, a Yao-graph partitions the plane around each vertex into m disjoint cones, each having aperture = 2p/m, and adds an edge to the closest vertex in each cone. Constrained Yao-graphs have the additional property that no edge properly intersects any of the given line segment constraints. Constrained -graphs are similar to constrained Yao-graphs, but use a different method to determine the closest vertex. We present tight bounds on the spanning ratio of a large family of constrained -graphs. We show that constrained -graphs with 4k + 2 (k ≥ 1 and integer) cones have a tight spanning ratio of 1 + 2sin(/2), where is 2p/(4k + 2). We also present improved upper bounds on the spanning ratio of the other families of constrained -graphs. These bounds match the current upper bounds in the unconstrained setting. We also show that constrained Yao-graphs with an even number of cones (m ≥ 8) have spanning ratio at most 1/(1 - 2sin(/2)) and constrained Yao-graphs with an odd number of cones (m ≥ 5) have spanning ratio at most 1/(1 - 2sin(3/8)). As is the case with constrained -graphs, these bounds match the current upper bounds in the unconstrained setting, which implies that like in the unconstrained setting using more cones can make the spanning ratio worse.

constraints, geometric graphs, Spanners, spanning ratio, tight bounds
dx.doi.org/10.1142/S021819591950002X
International Journal of Computational Geometry and Applications
School of Computer Science

Bose, P, & Van Renssen, A. (André). (2019). Spanning Properties of Yao and -Graphs in the Presence of Constraints. International Journal of Computational Geometry and Applications, 29(2), 95–120. doi:10.1142/S021819591950002X