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Lax17.TerminalElementMenger

Terminal element-Menger theorem

concepts/Lax17/TerminalElementMenger.lean · lax-17

proven

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    Theorem

    For a named-edge multigraph, the canonical available-boundary formulation of terminal element-connectivity is equivalent to the usual formulation by arbitrary terminal element cuts.

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    1import Lax17.TerminalConnectivity
    2
    3/-!
    4---
    5title: Terminal element-Menger theorem
    6type: theorem
    7---
    8For a named-edge multigraph, the canonical available-boundary formulation of
    9terminal element-connectivity is equivalent to the usual formulation by
    10arbitrary terminal element cuts.
    11-/
    12
    13namespace Lax17.TerminalElementMenger
    14
    15universe u
    16
    17/-- Terminal element-connectivity in arbitrary-cut form. -/
    18axiom terminalElementMenger :
    19 ∀ {V : Type u} [Fintype V] [DecidableEq V]
    20 (H : Lax17.TerminalConnectivity.EdgeIndexedGraph V)
    21 (terminals : Finset V) (k : ℕ),
    22 H.TerminalElementConnectedAtLeast terminals k ↔
    23 ∀ ⦃a : V⦄, a ∈ terminals →
    24 ∀ ⦃b : V⦄, b ∈ terminals → a ≠ b →
    25 ∀ C : H.ElementCut terminals a b, k ≤ C.order
    26
    27end Lax17.TerminalElementMenger
    28
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