Edge-connectivity of undirected and directed hypergraphs

84 Chapter 5. Partition-connectivity of hypergraphs
H = (V, E) is spanning if V = ∪(E ′ ). One can consider the problem of decomposing a
hypergraph into k connected spanning sub-hypergraphs, or the problem of decomposition
into k partition-connected spanning sub-hypergraphs. As the following theorem shows,
the former problem is NP-complete, while we will see that the latter is solvable and has
relatively simple structural properties.
Theorem 5.1. The problem of deciding whether a hypergraph H = (V, E) can be decom-
posed into k connected spanning sub-hypergraphs is NP-complete for every integer k ≥ 2.
Proof. First assume that k = 2. Recall that it is NP-complete to decide whether the nodes
of a hypergraph can be coloured by 2 colours such that every hyperedge contains nodes
of both colours. This implies by duality that colouring the hyperedges of a hypergraph
H0 = (V0, E0) by red and blue so that every node belongs to both a red and a blue hyperedge
is also NP-complete. We show that this latter problem is polynomially solvable if there
is a polynomial algorithm to decide decomposability of a hypergraph into two connected
spanning sub-hypergraphs. Let s be a new node, let V := V0 + s, E := {e + s : e ∈ E0},
and H = (V, E). Note that a sub-hypergraph H ′ = (V, E ′ ) of H is connected and spans
V if and only if the corresponding sub-hypergraph (V0, E ′ 0) of H0 spans V0. Therefore H
can be decomposed into 2 connected spanning sub-hypergraphs if and only if H0 can be
decomposed into two sub-hypergraphs both spanning V0.
The NP-completeness of the problem for k ≥ 3 easily reduces to the case k = 2. Let
H = (V, E) be a hypergraph and let H + denote the hypergraph arising from H by adding
k − 2 copies of V as new hyperedges. Then H + can be decomposed into k connected
spanning sub-hypergraphs if and only if H can be decomposed into 2 connected spanning
sub-hypergraphs.
5.1.2 k-partition-connectivity
A hypergraph H = (V, E) is called k-partition-connected if for every t, one has to delete at
least kt hyperedges to dismantle it into t+1 components. Equivalently, eH(F) ≥ k(|F|−1)
for every partition F of V . This is clearly a generalization of k-partition-connectivity for
graphs. In the graph case, the property was also definable in terms of decomposition
into connected spanning subgraphs (see Theorem 1.5). As an analogous result, we will
show that k-partition-connectivity of a hypergraph is equivalent to decomposability into k
partition-connected spanning sub-hypergraphs.
Another equivalent characterization of k-partition-connectivity of graphs is the existence
of k edge-disjoint spanning trees. The problem of finding k edge-disjoint spanning trees
in a graph is a special case of finding k disjoint bases of a matroid and therefore Theorem