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Distance to planar graphs

Lax825442.DistanceToPlanar · concepts/Lax825442/DistanceToPlanar.lean · lax-825442

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    Natural Language Statement

    Definition

    The distance to planar graphs is the minimum number of vertices whose deletion leaves a planar induced graph. Planarity uses the registered Lax68.Planar.IsPlanarLax68.Planar.IsPlanar, which asserts the existence of a crossing-free straight-line drawing in the real plane. For finite simple graphs this agrees with standard planarity.

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    Lean source view on GitHub

    1import Lax68.Planar
    2import Mathlib.Combinatorics.SimpleGraph.Finite
    3import Mathlib.Order.Lattice.Nat
    4
    5/-!
    6---
    7title: Distance to planar graphs
    8type: definition
    9---
    10The distance to planar graphs is the minimum number of vertices whose deletion
    11leaves a planar induced graph. Planarity uses the registered `Lax68.Planar.IsPlanar`,
    12which asserts the existence of a crossing-free straight-line drawing in the
    13real plane. For finite simple graphs this agrees with standard planarity.
    14-/
    15
    16namespace Lax825442.DistanceToPlanar
    17
    18/-- Minimum number of vertices whose deletion leaves a planar graph. -/
    19noncomputable def distanceToPlanar {V : Type} [Fintype V] [DecidableEq V]
    20 (G : SimpleGraph V) : ℕ :=
    21 sInf {k : ℕ | ∃ S : Finset V, (S.card = k) ∧
    22 (Lax68.Planar.IsPlanar (G.induce {v | v ∉ S}))}
    23
    24end Lax825442.DistanceToPlanar
    25
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