Linear-size planar Manhattan network for convex point sets

Let G=(V,E) be an edge weighted geometric graph (not necessarily planar) such that every is horizontal or vertical. The weight of uv∈E the L1-distance between its endpoints. WG(u,v) denotes length a shortest path pair vertices u and v in G. G said to Manhattan network for given point set P plane if P⊆V ∀p,q∈P, WG(p,q)=‖pq‖1. In addition P, may also include T Steiner points vertex V. problem, objective construct small size (the number points) n points. This problem was first considered by Gudmundsson et al. [EuroCG 2007]. They give construction Θ(nlog⁡n) general sets plane. We say planar it has embedding. this paper, we convex linear time using O(n) show that, even sets, 2007] needs Ω(nlog⁡n) points, not planar.