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Lax68.GraphMinors

Graph minors

concepts/Lax68/GraphMinors.lean · lax-68

definition

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    Definition

    A graph H is a minor of a graph G when the vertices of H can be represented by pairwise disjoint connected branch sets in G, with an edge joining the corresponding branch sets for every edge of H.

    Lean source view on GitHub

    1import Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected
    2
    3/-!
    4---
    5title: Graph minors
    6type: definition
    7---
    8A graph *H* is a minor of a graph *G* when the vertices of *H* can be
    9represented by pairwise disjoint connected branch sets in *G*, with an edge
    10joining the corresponding branch sets for every edge of *H*.
    11-/
    12
    13set_option autoImplicit false
    14
    15namespace Lax68.GraphMinors
    16
    17structure MinorModel {W V : Type*}
    18 (H : SimpleGraph W) (G : SimpleGraph V) where
    19 branchSet : W → Set V
    20 connected : ∀ w, (G.induce (branchSet w)).Connected
    21 disjoint :
    22 ∀ {u v}, u ≠ v →
    23 Disjoint (branchSet u) (branchSet v)
    24 adjacent :
    25 ∀ {u v}, H.Adj u v →
    26 ∃ x ∈ branchSet u, ∃ y ∈ branchSet v, G.Adj x y
    27
    28def IsMinor {W V : Type*}
    29 (H : SimpleGraph W) (G : SimpleGraph V) : Prop :=
    30 Nonempty (MinorModel H G)
    31
    32abbrev K5 : SimpleGraph (Fin 5) :=
    33 SimpleGraph.completeGraph (Fin 5)
    34
    35abbrev K33 : SimpleGraph (Fin 3 ⊕ Fin 3) :=
    36 completeBipartiteGraph (Fin 3) (Fin 3)
    37
    38end Lax68.GraphMinors
    39

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