Edge-connectivity of undirected and directed hypergraphs

12 Chapter 1. Introduction and preliminaries
a directed graph D = (V, A) whose underlying graph is G.
A set function is a function p : 2 V → Q ∪ {−∞, +∞} (sometimes the set function
is defined only on a given family of subsets of V ). It is assumed that p(∅) = 0, unless
otherwise stated. For a function m : V → Q and a set X ⊆ V , we use the notation
m(X) :=
m(v).
v∈X
A graph G is said to cover a set function p if dG(X) ≥ p(X) for every X ⊆ V . Anal-
ogously, a digraph D covers p if ϱD(X) ≥ p(X) for every X ⊆ V . Given a function
m : V → Z+, the graph G satisfies the degree-specification m if dG(v) = m(v) for every
v ∈ V . If we have an in-degree specification mi : V → Z+ and an out-degree specifica-
tion mo : V → Z+, the digraph D satisfies the degree-specifications if ϱD(v) = mi(v) and
δD(v) = mo(v) for every v ∈ V .
In optimization problems we often consider a cost function c : E → Q on the edges of
the graph (G = V, E) or c : A → Q on the edges of a digraph D = (V, A). A cost function
on E is said to be node induced if c(uv) = c ′ (u) + c ′ (v) where c ′ : V → Q is a cost function
on the nodes. Similarly, a cost function c on A is node induced if c(uv) = c ′ o(u) + c ′ i(v)
where c ′ i : V → Q and c ′ o : V → Q are cost functions on the nodes.
1.2.2 The notion of edge-connectivity
For two nodes s ∈ V and t ∈ V , a set X ⊆ V is called an st-set if s /∈ X and t ∈ X.
Description of undirected and directed edge-connectivity is based on the following versions
of Menger’s Theorem (see e.g. [21]):
Theorem 1.1 (Menger). Let G = (V, E) be a graph, and s, t ∈ V distinct nodes. The
maximum number of edge-disjoint paths between s and t is
min{dG(X) : X ⊆ V is an st-set}.
Theorem 1.2 (Menger). Let D = (V, A) be a digraph, and s, t ∈ V distinct nodes. The
maximum number of edge-disjoint directed paths from s to t is
min{ϱD(X) : X ⊆ V is an st-set}.
We use the notation λG(s, t) for the maximum number of edge-disjoint paths between s
and t in G, and λD(s, t) for the maximum number of edge-disjoint directed paths from s
to t in D. These values are called the local edge-connectivity between s and t (from s to t).
The fact that the global edge-connectivity of a graph is high can be formulated in different
ways: