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_5: We characterize the planar straight line graphs (Pslgs) that can be augmented to 3-connected and 3-edge-connected Pslgs, respectively. We show that if a Pslg with n vertices can be augmented to a 3-edge-connected Pslg, then at most 2n−2 new edges are always sufficient and sometimes necessary for the augmentation. If the input Pslg is, in addition, already 2-edge-connected, then n−2 new edges are always sufficient and sometimes necessary for the augmentation to a 3-edge-connected Pslg.
  @cite_8: Abstract Let S be a set of n ⩾ 4 points in general position in the plane, and let h n be the number of extreme points of S . We show how to construct a 3-connected plane graph with vertex set S , having max ⌈ 3 n 2 ⌉ , n + h − 1 edges, and we prove that there is no 3-connected plane graph on top of S with a smaller number of edges. In particular, this implies that S admits a 3-connected cubic plane graph if and only if n ⩾ 4 is even and h ⩽ n 2 + 1 . The same bounds also hold when 3-edge-connectivity is considered. We also give a partial characterization of the point sets in the plane that can be the vertex set of a cubic plane graph.
  @cite_11: It is shown that if a planar straight line graph (Pslg) with n vertices in general position in the plane can be augmented to a 3-edge-connected Pslg, then 2n ? 2 new edges are enough for the augmentation. This bound is tight: there are Pslgs with n ? 4 vertices such that any augmentation to a 3-edge-connected Pslg requires 2n ? 2 new edges.
summary: T 'oth and Valter @cite_5 characterize geometric planar graphs that can be augmented to 3-edge connected planar graphs. Later Al- @cite_11 gave a tight upper bound on the number of added edges in such augmentations. Finally, Garc ' @cite_8 show how to construct a 3-connected geometric planar graph on a set of points in the planar with the minimum number of straight line edges of unbounded length.