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Topological graph minors

Lax68.GraphTopologicalMinors · concepts/Lax68/GraphTopologicalMinors.lean · lax-68

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    Natural Language Statement

    Definition

    Topological minor illustration
    Topological minor illustration

    A graph H is a topological minor of G when the vertices of H are represented by distinct branch vertices of G and its edges by paths whose interiors contain no branch vertex and are pairwise disjoint. Equivalently, G contains a subdivision of H as a subgraph.

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    In the paper

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    1import Mathlib.Combinatorics.SimpleGraph.Paths
    2import Lax68.GraphMinors
    3
    4/-!
    5---
    6title: Topological graph minors
    7type: definition
    8---
    9
    10![Topological minor illustration](https://raw.githubusercontent.com/lax-archive/lax-submissions/4dd917c8b9a1e181ec951b7f6fa384262af31fa9/planar-graph-classes-v4-33/assets/topological-minor.svg "Topological minor illustration")
    11
    12A graph *H* is a topological minor of *G* when the vertices of *H* are
    13represented by distinct branch vertices of *G* and its edges by paths whose
    14interiors contain no branch vertex and are pairwise disjoint. Equivalently,
    15*G* contains a subdivision of *H* as a subgraph.
    16-/
    17
    18set_option autoImplicit false
    19
    20namespace Lax68.GraphTopologicalMinors
    21
    22def walkInterior {V : Type*} {G : SimpleGraph V} {a b : V}
    23 (P : G.Walk a b) : Set V :=
    24 {x | x ∈ P.support ∧ x ≠ a ∧ x ≠ b}
    25
    26structure TopologicalMinorModel {W V : Type*}
    27 (H : SimpleGraph W) (G : SimpleGraph V) where
    28 branch : W ↪ V
    29 route :
    30 ∀ {a b : W}, H.Adj a b →
    31 G.Walk (branch a) (branch b)
    32 route_isPath :
    33 ∀ {a b : W} (h : H.Adj a b),
    34 (route h).IsPath
    35 branch_avoids_interiors :
    36 ∀ {a b : W} (h : H.Adj a b) (w : W),
    37 branch w ∉ walkInterior (route h)
    38 route_interiors_disjoint :
    39 ∀ {a b c d : W}
    40 (hab : H.Adj a b) (hcd : H.Adj c d),
    41 ¬ ((a = c ∧ b = d) ∨ (a = d ∧ b = c)) →
    42 Disjoint
    43 (walkInterior (route hab))
    44 (walkInterior (route hcd))
    45
    46def IsTopologicalMinor {W V : Type*}
    47 (H : SimpleGraph W) (G : SimpleGraph V) : Prop :=
    48 Nonempty (TopologicalMinorModel H G)
    49
    50def IsKuratowskiFree {V : Type*} (G : SimpleGraph V) : Prop :=
    51 ¬IsTopologicalMinor Lax68.GraphMinors.K5 G ∧
    52 ¬IsTopologicalMinor Lax68.GraphMinors.K33 G
    53
    54end Lax68.GraphTopologicalMinors
    55
    Builds on
    Used by
    From Mathlib

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