Let
![$G=(V+s,E)$](Abs/img1.gif)
be a graph and let
![${\cal S}=(d_1,...,d_p)$](Abs/img2.gif)
be
a set of positive integers with
![$\sum d_j=d(s)$](Abs/img3.gif)
. An
![$\cal S$](Abs/img4.gif)
-detachment
splits
![$s$](Abs/img5.gif)
into a set of
![$p$](Abs/img6.gif)
independent vertices
![$s_1,...,s_p$](Abs/img7.gif)
with
![$d(s_j)=d_j$](Abs/img8.gif)
,
![$1\leq j\leq p$](Abs/img9.gif)
. Given a requirement function
![$r(u,v)$](Abs/img10.gif)
on pairs
of vertices of
![$V$](Abs/img11.gif)
, an
![$\cal S$](Abs/img4.gif)
-detachment is called
![$r$](Abs/img12.gif)
-admissible if the
detached graph
![$G'$](Abs/img13.gif)
satisfies
![$\lambda_{G'}(x,y)\geq r(x,y)$](Abs/img14.gif)
for every pair
![$x,y\in V$](Abs/img15.gif)
. Here
![$\lambda_H(u,v)$](Abs/img16.gif)
denotes the local edge-connectivity between
![$u$](Abs/img17.gif)
and
![$v$](Abs/img18.gif)
in graph
![$H$](Abs/img19.gif)
.
We prove that an
-admissible
-detachment exists if and only if (a)
, and (b)
hold for every
.
The special case of this characterization when
for each pair in
was conjectured by B. Fleiner.
Our result is a common generalization of a theorem of W. Mader on edge
splittings preserving local edge-connectivity and a result of B. Fleiner on
detachments preserving global edge-connectivity. Other corollaries include
previous results of L. Lovász and C. J. St. A. Nash-Williams on edge
splittings and detachments, respectively. As a new application, we extend a
theorem of A. Frank on local edge-connectivity augmentation to the case when
stars of given degrees are added