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Feedback vertex number

Lax379983.FeedbackVertexNumber · concepts/Lax379983/FeedbackVertexNumber.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 vertex set is a set S⊆VS\subseteq V for which the induced graph G[V∖S]G[V\setminus S] is acyclic. The feedback vertex number is

    FVN⁡(G)=min⁡{∣S∣:S⊆V,  G[V∖S] is acyclic}.\operatorname{FVN}(G)=\min\{\lvert S\rvert : S\subseteq V,\; G[V\setminus S]\text{ is acyclic}\}.

    The minimum exists because deleting all vertices 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 vertex number
    8type: definition
    9---
    10Let G=(V,E)G=(V,E) be a finite undirected simple graph. A feedback vertex set is a
    11set S⊆VS\subseteq V for which the induced graph G[V∖S]G[V\setminus S] is acyclic.
    12The feedback vertex number is
    13
    14FVN⁡(G)=min⁡{∣S∣:S⊆V,  G[V∖S] is acyclic}.\operatorname{FVN}(G)=\min\{\lvert S\rvert : S\subseteq V,\; G[V\setminus S]\text{ is acyclic}\}.
    15
    16
    17The minimum exists because deleting all vertices leaves an acyclic graph.
    18-/
    19
    20namespace Lax379983.FeedbackVertexNumber
    21
    22/-- The minimum number of vertices whose deletion makes the graph acyclic. -/
    23noncomputable def feedbackVertexNumber {V : Type*} [Finite V] (G : SimpleGraph V) : ℕ :=
    24 sInf {n : ℕ | ∃ S : Set V, S.ncard = n ∧ (G.induce Sᶜ).IsAcyclic}
    25
    26end Lax379983.FeedbackVertexNumber
    27
    Builds on

    none

    Used by
    From Mathlib

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