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Feedback edge number

Lax379983.FeedbackEdgeNumber · concepts/Lax379983/FeedbackEdgeNumber.lean · lax-379983

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    Natural Language Statement

    Definition

    Let G=(V,E)G=(V,E) be a finite undirected simple graph. A feedback edge set is a set F⊆EF\subseteq E for which the graph G−FG-F, obtained by deleting the edges in FF and retaining all vertices, is acyclic. The feedback edge number is

    FEN⁡(G)=min⁡{∣F∣:F⊆E,  G−F is acyclic}.\operatorname{FEN}(G)=\min\{\lvert F\rvert : F\subseteq E,\; G-F\text{ is acyclic}\}.

    The minimum exists because deleting all edges leaves an acyclic graph.

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    Lean source view on GitHub

    1import Mathlib.Combinatorics.SimpleGraph.Acyclic
    2import Mathlib.Data.Nat.Lattice
    3import Mathlib.Data.Set.Card
    4
    5/-!
    6---
    7title: Feedback edge number
    8type: definition
    9---
    10Let G=(V,E)G=(V,E) be a finite undirected simple graph. A feedback edge set is a
    11set F⊆EF\subseteq E for which the graph G−FG-F, obtained by deleting the edges
    12in FF and retaining all vertices, is acyclic. The feedback edge number is
    13
    14FEN⁡(G)=min⁡{∣F∣:F⊆E,  G−F is acyclic}.\operatorname{FEN}(G)=\min\{\lvert F\rvert : F\subseteq E,\; G-F\text{ is acyclic}\}.
    15
    16
    17The minimum exists because deleting all edges leaves an acyclic graph.
    18-/
    19
    20namespace Lax379983.FeedbackEdgeNumber
    21
    22/-- The minimum number of edges whose deletion makes the graph acyclic. -/
    23noncomputable def feedbackEdgeNumber {V : Type*} [Finite V] (G : SimpleGraph V) : ℕ :=
    24 sInf {n : ℕ | ∃ F : Set (Sym2 V),
    25 F ⊆ G.edgeSet ∧ F.ncard = n ∧ (G.deleteEdges F).IsAcyclic}
    26
    27end Lax379983.FeedbackEdgeNumber
    28
    Builds on

    none

    Used by
    From Mathlib

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