Edge-connectivity of undirected and directed hypergraphs

Section 6.2. Orientation of graphs 99
flow problem given by a crossing supermodular set function involves the full truncation of
the set function.
6.2.3 Constructive characterizations
One important application of the connection between the connectivity properties of undi-
rected graphs and the connectivity properties of their orientations concerns the theory of
constructive characterizations. Constructive characterization of a connectivity property
means that such (di)graphs can be constructed from a small initial (di)graph by some
simple operations that preserve the property.
For (k, l)-edge-connectivity of digraphs, the following conjecture is expected to be true:
Conjecture 6.6. Let k > l be non-negative integers. A digraph is (k, l)-edge-connected if
and only if it can be built from a single node by the following two operations:
(i) Add a new edge to the digraph,
(ii) Pinch i (l ≤ i < k) existing edges with a new node z, and add k−i new edges entering
z and leaving existing nodes,
where pinching edges u1v1, . . . , utvt with z means deleting those edges, and adding the edges
u1z, zv1, u2z, zv2, . . . , utz, zvt.
It is easy to see that the above operations create a (k, l)-edge-connected digraph. The
conjecture was proved for l = 1 by Frank and Szegő [34], and for l = k − 1 by Frank and
Z. Király [31]. It is also easy to see that since for k > l (k, l)-partition-connected graphs are
exactly those that have a (k, l)-edge-connected orientation, proof of this conjecture would
also imply the following on the constructive characterization of (k, l)-partition-connected
graphs:
Conjecture 6.7. Let k > l be non-negative integers. A graph is (k, l)-partition-connected
if and only if it can be built from a single node by the following two operations:
(i) Add a new edge to the graph,
(ii) Pinch i (l ≤ i < k) existing edges with a new node z, and add k − i new edges
connecting z with existing nodes.