We introduce a class of thin combings and prove that it has a nice behaviour in free constructions. Among corollaries, we deduce the relative metabolicity of Sela’s limit groups. S. Gersten defined the property of metabolicity for finitely presented groups (or even, contractible spaces with bounded geometry in dimension 2) as an attempt of an algebraic approach of word-hyperbolic groups [8] [9]...

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.

It is shown that every connected planar straight line graph with n ≥ 3 vertices has an embedding preserving augmentation to a 2-edge connected planar straight line graph with at most b(2n − 2)/3c new edges. It is also shown that every planar straight line tree with n ≥ 3 vertices has an embedding preserving augmentation to a 2-edge connected planar topological graph by adding at most bn/2c edge...