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Box representations

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

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    Natural Language Statement

    Definition

    A box representation assigns a nonempty closed axis-parallel real box to each vertex, with adjacency exactly when the boxes intersect. The dimension may be zero, in which case the represented graph is complete.

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    1import Mathlib.Combinatorics.SimpleGraph.Basic
    2import Mathlib.Data.Real.Basic
    3
    4/-!
    5---
    6title: Box representations
    7type: definition
    8---
    9A box representation assigns a nonempty closed axis-parallel real box to
    10each vertex, with adjacency exactly when the boxes intersect. The dimension
    11may be zero, in which case the represented graph is complete.
    12-/
    13
    14namespace Lax560391.BoxRepresentation
    15
    16structure BoxRepresentation {V : Type} (G : SimpleGraph V) (k : ℕ) where
    17 lo : V → Fin k → ℝ
    18 hi : V → Fin k → ℝ
    19 nonempty : ∀ v i, lo v i ≤ hi v i
    20 represents : ∀ u v, u ≠ v →
    21 (G.Adj u v ↔ ∀ i, lo u i ≤ hi v i ∧ lo v i ≤ hi u i)
    22
    23end Lax560391.BoxRepresentation
    24
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    From Mathlib

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