Some Properties of Continuous Yao Graph

摘要

Given a set S of points in the plane and an angle 0 ＜ θ ≤ 2π, the continuous Yao graph cY(θ) with vertex set S and angle θ defined as follows. For each p,q ∈ S, we add an edge from p to q in cY(θ) if there exists a cone with apex p and angular diameter θ such that q is the closest point to p inside this cone. In this paper, we prove that for 0 ＜ θ ＜ π/3 and t ≥ 1/(1-2sin(θ/2)), the continuous Yao graph cY(θ) is a C-fault-tolerant geometric t-spanner where C is the family of convex regions in the plane. Moreover, we show that for every θ ≤ π and every half-plane h, cY(θ) (Θ) h is connected, where cY(θ) (Θ) h is the graph after removing all edges and points inside h from the graph cY(θ). Also, we show that there is a set of n points in the plane and a convex region C such that for every θ ≥ π/3, cY(θ) (Θ) C is not connected. Given a geometric network G and two vertices x and y of G, we call a path P from x to y a self-approaching path, if for any point q on P, when a point p moves continuously along the path from x to q, it always get closer to q. A geometric graph G is self-approaching, if for every pair of vertices x and y there exists a self-approaching path in G from x to y. In this paper, we show that there is a set P of n points in the plane such that for some angles θ, Yao graph on P with parameter θ is not a self-approaching graph. Instead, the corresponding continuous Yao graph on P is a self-approaching graph. Furthermore, in general, we show that for every θ ＞ 0, cY(θ) is not necessarily a self-approaching graph.