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Vertex-Menger theorem

Lax17.VertexMenger · concepts/Lax17/VertexMenger.lean · lax-17

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    Natural Language Statement

    Theorem

    The vertex form of Menger's theorem gives an exact alternative between kk vertex-disjoint terminal-to-terminal paths and a separator of size less than kk.

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
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    1import Lax17.Paths
    2
    3/-!
    4---
    5title: Vertex-Menger theorem
    6type: theorem
    7---
    8The vertex form of Menger's theorem gives an exact alternative between
    9\(k\) vertex-disjoint terminal-to-terminal paths and a separator of size
    10less than \(k\).
    11-/
    12
    13namespace Lax17.VertexMenger
    14
    15universe u
    16
    17/-- Finite vertex-Menger in packing-or-separator form. -/
    18axiom vertexMenger :
    19 ∀ {V : Type u} [Fintype V] [DecidableEq V]
    20 (G : SimpleGraph V) (A B : Finset V) (k : ℕ),
    21 Nonempty (Lax17.Paths.VertexLinkage G A B k) ∨
    22 ∃ X : Finset V,
    23 X.card < k ∧ Lax17.Paths.IsVertexSeparator G A B X
    24
    25end Lax17.VertexMenger
    26
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