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Treewidth is at most feedback vertex number plus one

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

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    Natural Language Statement

    Theorem

    For every finite undirected simple graph GG,

    tw⁡(G)≤FVN⁡(G)+1.\operatorname{tw}(G)\leq\operatorname{FVN}(G)+1.

    Here treewidth is the least ww for which a tree decomposition exists with every bag of size at most w+1w+1. The bound also holds for empty and disconnected graphs.

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

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    Lean source view on GitHub

    1import Lax228581.Treewidth
    2import Lax379983.FeedbackVertexNumber
    3
    4/-!
    5---
    6title: Treewidth is at most feedback vertex number plus one
    7type: theorem
    8---
    9For every finite undirected simple graph GG,
    10
    11tw⁡(G)≤FVN⁡(G)+1.\operatorname{tw}(G)\leq\operatorname{FVN}(G)+1.
    12
    13Here treewidth is the least ww for which a tree decomposition exists with
    14every bag of size at most w+1w+1. The bound also holds for empty and
    15disconnected graphs.
    16-/
    17
    18namespace Lax379983.TreewidthBound
    19
    20open Lax228581.Treewidth Lax379983.FeedbackVertexNumber
    21
    22/-- Every finite simple graph has treewidth at most its feedback vertex number plus one. -/
    23axiom treewidth_le_feedbackVertexNumber_add_one {V : Type} [Fintype V] [DecidableEq V]
    24 (G : SimpleGraph V) : treewidth G ≤ feedbackVertexNumber G + 1
    25
    26end Lax379983.TreewidthBound
    27
    Show Proof
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