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Lax68.Planar

Planar graphs

concepts/Lax68/Planar.lean · lax-68

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    Definition

    A graph is planar when it can be drawn in the plane without crossings. For finite graphs, this is equivalent to containing neither the complete graph K₅ nor the complete bipartite graph K₃,₃ as a minor. The drawing certificate and graph-minor relation are supplied by separate concepts.

    Lean source view on GitHub

    1import Lax68.StraightLineDrawings
    2import Lax68.GraphMinors
    3
    4/-!
    5---
    6title: Planar graphs
    7type: definition
    8---
    9A graph is planar when it can be drawn in the plane without crossings.
    10For finite graphs, this is equivalent to containing neither the complete graph
    11*K₅* nor the complete bipartite graph *K₃,₃* as a minor. The drawing certificate
    12and graph-minor relation are supplied by separate concepts.
    13-/
    14
    15set_option autoImplicit false
    16
    17namespace Lax68.Planar
    18
    19open GraphMinors
    20
    21def IsPlanar {V : Type*} (G : SimpleGraph V) : Prop :=
    22 StraightLineDrawings.HasStraightLineDrawing G
    23
    24def IsPlanarByExcludedMinors {V : Type*} (G : SimpleGraph V) : Prop :=
    25 ¬IsMinor K5 G ∧
    26 ¬IsMinor K33 G
    27
    28end Lax68.Planar
    29

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