Edge-connectivity of undirected and directed hypergraphs

Notation
Z+
the set of non-negative integers
(x) + for x ∈ R: max{x, 0}
x > 0 for x ∈ R n : x ≥ 0, x = 0
x ≫ 0 for x ∈ R n : xi > 0 (i = 1, . . . , n)
X ⊆ V X is a subset of V (possibly X = V )
X ⊂ V X is a proper subset of V
X − Y for X, Y ⊆ V : {v ∈ V : v ∈ X, v /∈ Y }
χX
m(X)
for X ⊆ V : the characteristic function of the set X
for X ⊆ V and m : V → R:
v∈X m(v)
dH(v) for a hypergraph H = (V, E) and v ∈ V : the degree of v in H
ϱD(v) for a dir. hypergraph D = (V, A): the in-degree of v in D
δD(v) the out-degree of v in D
∆H(X) {e ∈ E : e enters X}
∆ −
D (X) {a ∈ A : a enters X}
∆ +
D
(X) {a ∈ A : a enters V − X}
dH(X) |∆H(X)|
dE(X) dH(X) for H = (V, E)
ϱD(X) |∆ −
D (X)|
δD(X) |∆ +
D (X)|
iH(X) the number of hyperedges of H induced by X
ϱx(Z) for D = (V, A) and x : A → R:
φx(Z) δx(Z) − ϱx(Z)
a∈∆ −
D
st-set a set X ⊆ V for which s /∈ X and t ∈ X
(Z) x(a)
λH(s, t) local edge-connectivity between s and t in H
λD(s, t) local edge-connectivity from s to t in D
χe(v) for v ∈ V and a hyperedge e: the multiplicity of v in e
|e ∩ X| for X ⊆ V and a hyperedge e: χe(X)
∪(E) for H = (V, E): {v ∈ V : χe(v) > 0 for some e ∈ E}
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