  The `lifting` or `splitting-off` operation on graphs is performed by deleting
two edges sv and sw having a common end s and adding a new edge between v and
w. Such a lift is considered good if it preserves a certain local
edge-connectivity between the pairs of vertices different from the vertex s at
which lifting takes place. The operation is important for inductive proofs
concerning edge-connectivity, and can be seen widely applied in the literature
on connectivity augmentation, network design, orientation (of finite and
infinite graphs), and edge-disjoint linkage. It was studied by Lovasz, who used
the term splitting-off, and Mader, who used the term lifting. They proved the
first two significant results on it, in 1976 and 1978 respectively, showing the
existence of a good lift under certain conditions. Then it was used and studied
by other researchers, through the 1980s, both for undirected and directed
graphs. In particular, it was investigated further by Frank who proved in 1992
that there are floor of deg(s)/2 disjoint good lifts. Motivated by the
applications, a new method for studying the operation was introduced by Jordan
in the late 1990s. He defined and studied the structure of the
`non-admissibility` graph, which is the complement of the lifting graph; the
subject of this paper. He proved a number of significant structural results on
it, which he applied to connectivity augmentation. Independently, in 2016,
Thomassen defined the `lifting graph`, and called its complement the `bad
graph`, to apply it in finding orientations of infinite graphs. Later in the
same year, Thomassen with Ok and Richter extended the study, and applied their
results to linkages in infinite graphs.
  Here we give a more comprehensive analysis of the structure of the lifting
graph.
