Edge-connectivity of undirected and directed hypergraphs

14 Chapter 1. Introduction and preliminaries
contains a spanning tree, and a digraph is strongly connected if and only if it contains a
spanning arborescence rooted at s for every s ∈ V .
Given a graph G = (V, E) and a subset W ⊆ V , a subtree G ′ = (V ′ , E ′ ) of G is called
a Steiner tree for W if W ⊆ V ′ . A graph is connected in W if and only if it contains a
Steiner tree for W .
In [70] Tutte investigated the problem of decomposing a graph into a given number
of connected spanning subgraphs, which is equivalent to finding a given number of edge-
disjoint spanning trees of G. He proved the following fundamental result:
Theorem 1.5 (Tutte). An undirected graph G = (V, E) contains k edge-disjoint span-
ning trees (or G can be decomposed into k connected spanning subgraphs) if and only if
eG(F) ≥ k(|F| − 1) (1.1)
holds for every partition F of V into non-empty subsets, where eG(F) denotes the number
of edges connecting distinct members of F.
Note that given a graph G = (V, E) and a subset W ⊆ V , it is NP-complete to decide
whether G contains k edge-disjoint Steiner trees for W . This problem will be discussed in
greater detail in Chapter 5.
The following result of Nash-Williams [63] is in some sense a complementary pair of
Tutte’s theorem:
Theorem 1.6 (Nash-Williams). A graph G can be covered by k forests if and only if
iG(X) ≤ k(|X| − 1) for every non-empty subset X of V .
For digraphs, the following result of Edmonds [14] is an analogue of Tutte’s theorem:
Theorem 1.7 (Edmonds). Let D = (V, A) be a digraph, s ∈ V a fixed root node. Then
D contains k edge-disjoint spanning arborescences rooted at s if and only if
ϱD(X) ≥ k for every ∅ = X ⊆ V − s.
In fact, Edmonds stated the result in a more general form:
Theorem 1.8 (Edmonds). Let D = (V, A) be a digraph, and S1, . . . , Sk subsets of V ;
for X ⊆ V , let f(X) denote the number of sets Si not disjoint from X. Then D can be
decomposed into directed subgraphs D1, . . . , Dk such that Di is connected from Si if and
only if
ϱD(X) ≥ k − f(X) for every ∅ = X ⊆ V .