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Chordal intersection dimension

Lax560391.ChordalIntersectionDimension · concepts/Lax560391/ChordalIntersectionDimension.lean · lax-560391

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    Natural Language Statement

    Definition

    The minimum number of chordal supergraphs whose edge intersection is the graph. The intersection of zero factors is complete, so complete graphs have value zero.

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    1import Lax560391.ChordalIntersectionRepresentation
    2import Mathlib.Order.Lattice.Nat
    3
    4/-!
    5---
    6title: Chordal intersection dimension
    7type: definition
    8---
    9The minimum number of chordal supergraphs whose edge intersection is the
    10graph. The intersection of zero factors is complete, so complete graphs have
    11value zero.
    12-/
    13
    14namespace Lax560391.ChordalIntersectionDimension
    15
    16/-- `G` is the intersection of `k` chordal graphs on the same vertices.
    17The distinctness guard gives the intersection of zero graphs the complete
    18graph rather than a reflexive relation. -/
    19def HasChordalIntersectionDimensionAtMost {V : Type}
    20 (G : SimpleGraph V) (k : ℕ) : Prop :=
    21 Nonempty (Lax560391.ChordalIntersectionRepresentation.ChordalIntersectionRepresentation G k)
    22
    23noncomputable def chordalIntersectionDimension
    24 {V : Type} [Fintype V] [DecidableEq V]
    25 (G : SimpleGraph V) : ℕ :=
    26 sInf {k | HasChordalIntersectionDimensionAtMost G k}
    27
    28end Lax560391.ChordalIntersectionDimension
    29
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