Approximating the Edge Length of 2-Edge Connected Planar Geometric Graphs on a Set of Points

Abstract

Given a set P of n points in the plane, we solve the problems of constructing a geometric planar graph spanning P 1) of minimum degree 2, and 2) which is 2-edge connected, respectively, and has max edge length bounded by a factor of 2 times the optimal; we also show that the factor 2 is best possible given appropriate connectivity conditions on the set P, respectively. First, we construct in O(nlogn) time a geometric planar graph of minimum degree 2 and max edge length bounded by 2 times the optimal. This is then used to construct in O(nlogn) time a 2-edge connected geometric planar graph spanning P with max edge length bounded by $\sqrt{5}$ times the optimal, assuming that the set P forms a connected Unit Disk Graph. Second, we prove that 2 times the optimal is always sufficient if the set of points forms a 2 edge connected Unit Disk Graph and give an algorithm that runs in O(n ²) time. We also show that for $k \in O(\sqrt{n})$, there exists a set P of n points in the plane such that even though the Unit Disk Graph spanning P is k-vertex connected, there is no 2-edge connected geometric planar graph spanning P even if the length of its edges is allowed to be up to 17/16.