Edge-connectivity of undirected and directed hypergraphs

28 Chapter 2. Submodular functions
V , F
ϕ : V → W
W , T = (W, A)
Figure 2.1: Tree-representation of a cross-free family F (the non-rounded rectangles rep-
resent their complement)
X
V , F
ϕ : V → W
W , T = (W, A)
ϕ(X)
ϕ(V − X)
Figure 2.2: A tree-composition F of X and its tree-representation (the non-rounded rect-
angles represent their complement)
A tree-composition F of a set X ⊆ V is a cross-free composition of X which contains
no partition or co-partition as a proper sub-family (so F itself can be a partition or a co-
partition, and it can also be empty). The name comes from the property that if ∅ = X ⊂ V ,
then F has a tree-representation (T = (W, A), ϕ) such that ϕ −1 (w) = ∅ for every w ∈ W (as
on Figure 2.2). In this tree-representation, ϱT (w) = 0 if ϕ −1 (w) ⊆ V − X, and δT (w) = 0
if ϕ −1 (w) ⊆ X. This implies the following, if we consider a node of W of the latter type
that is entered by more than one edge of T :
Claim 2.2. If F = ∅ is a tree-composition of X that is not a partition of X, then it
contains a subfamily {Z1, . . . , Zt} (t ≥ 2) of pairwise co-disjoint sets such that ∩Zi ⊆ X.
If X = V , then Zi − X = ∅ (i = 1, . . . , t).
Cross-free families can be decomposed into tree-compositions:
Lemma 2.3. Let the function c : V → Z+ have maximum value k, and let F be a cross-
free c-composition. Let Zi := {v ∈ V : c(v) ≥ i} (i = 1, . . . , k). Then F decomposes