text: Given a set @math of @math points in the plane, we solve the problems of constructing a geometric planar graph spanning @math 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 @math , respectively. First, we construct in @math 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 @math time a 2-edge connected geometric planar graph spanning @math with max edge length bounded by @math times the optimal, assuming that the set @math 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 @math time. We also show that for @math , there exists a set @math of @math points in the plane such that even though the Unit Disk Graph spanning @math is @math -vertex connected, there is no 2-edge connected geometric planar graph spanning @math even if the length of its edges is allowed to be up to 17 16.
  @cite_3: We give a ( 17 12 ) -approximation algorithm for the following NP- hard problem: Given a simple undirected graph, find a 2-edge connected span- ning subgraph that has the minimum number of edges. The best previous approximation guarantee was ( 3 2 ) . If the well known TSP ( 4 3 ) conjecture holds, then there is a ( 4 3 ) -approximation algorithm. Thus our main result gets half-way to this target.
  @cite_1: We present factor 4 3 approximation algorithms for the problems of finding theminimum 2-edge connected and theminimum 2-vertex connected spanning subgraph of a given undirected graph.
  @cite_12: 
  @cite_4: A 5 4-approximation algorithm is presented for the minimum cardinality 2-edge-connected spanning subgraph problem in undirected graphs. This improves the previous best approximation ratio of 4 3. It is shown that our ratio is tight with respect to current lower bounds, and any further improvement is possible only if new lower bounds are discovered.
summary: Our problem is also related to the well-known bottleneck traveling salesman problem, i.e. finding a Hamiltonian cycle that minimizes the length of the longest edge, since such a cycle is 2 edge conected (but not necessarily planar). @cite_12 gave a 2-approximation algorithm for this problem and also showed that there is no better algorithm unless @math . There is also literature on constructing @math edge connected subgraphs with minimum number of edges. In @cite_3 it is proved that given a @math -edge connected graph there is an algorithm running in time @math which finds a @math -edge connected spanning subgraph whose number of edges is @math times the optimal, where @math is the number of edges and @math the number of vertices of the graph. An improvement is provided in @cite_1 in which a 4 3 approximation algorithm is given. Later, @cite_4 provided a 5 4-approximation algorithm. However in these results the resulting spanning subgraphs are not guaranteed to be planar.