  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 $O(n\log{n})$ 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 $O(n\log n)$ 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.
