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Boxicity

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

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    Natural Language Statement

    Definition

    Boxicity is the minimum number of coordinates in an intersection representation by nonempty axis-parallel closed boxes. Zero coordinates give a complete graph, so complete graphs have value zero.

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    DefinitionThis conceptRelated conceptA → B: B builds on A

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    1import Lax560391.BoxRepresentation
    2import Mathlib.Order.Lattice.Nat
    3
    4/-!
    5---
    6title: Boxicity
    7type: definition
    8---
    9Boxicity is the minimum number of coordinates in an intersection
    10representation by nonempty axis-parallel closed boxes. Zero coordinates give a
    11complete graph, so complete graphs have value zero.
    12-/
    13
    14namespace Lax560391.Boxicity
    15
    16/-- An intersection representation by nonempty closed `k`-dimensional
    17axis-parallel boxes. -/
    18def HasBoxicityAtMost {V : Type} (G : SimpleGraph V) (k : ℕ) : Prop :=
    19 Nonempty (Lax560391.BoxRepresentation.BoxRepresentation G k)
    20
    21noncomputable def boxicity {V : Type} [Fintype V] [DecidableEq V]
    22 (G : SimpleGraph V) : ℕ :=
    23 sInf {k | HasBoxicityAtMost G k}
    24
    25end Lax560391.Boxicity
    26
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