Documentation

Mathlib.Combinatorics.SimpleGraph.Dart

Darts in graphs #

A Dart or half-edge or bond in a graph is an ordered pair of adjacent vertices, regarded as an oriented edge. This file defines darts and proves some of their basic properties.

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structure SimpleGraph.Dart {V : Type u_1} (G : SimpleGraph V) extends V × V :
Type u_1

A Dart is an oriented edge, implemented as an ordered pair of adjacent vertices. This terminology comes from combinatorial maps, and they are also known as "half-edges" or "bonds."

  • fst : V
  • snd : V
  • adj : G.Adj self.toProd.1 self.toProd.2
Instances For
    source
    instance SimpleGraph.instDecidableEqDart {V✝ : Type u_2} {G✝ : SimpleGraph V✝} [DecidableEq V✝] :
    DecidableEq G✝.Dart
    Equations
    • SimpleGraph.instDecidableEqDart = SimpleGraph.decEqDart✝
    source
    theorem SimpleGraph.Dart.ext_iff {V : Type u_1} {G : SimpleGraph V} (d₁ d₂ : G.Dart) :
    d₁ = d₂ ↔ d₁.toProd = d₂.toProd
    source
    theorem SimpleGraph.Dart.ext {V : Type u_1} {G : SimpleGraph V} (d₁ d₂ : G.Dart) (h : d₁.toProd = d₂.toProd) :
    d₁ = d₂
    source
    @[simp]
    theorem SimpleGraph.Dart.fst_ne_snd {V : Type u_1} {G : SimpleGraph V} (d : G.Dart) :
    d.toProd.1 ≠ d.toProd.2
    source
    @[simp]
    theorem SimpleGraph.Dart.snd_ne_fst {V : Type u_1} {G : SimpleGraph V} (d : G.Dart) :
    d.toProd.2 ≠ d.toProd.1
    source
    theorem SimpleGraph.Dart.toProd_injective {V : Type u_1} {G : SimpleGraph V} :
    Function.Injective SimpleGraph.Dart.toProd
    source
    instance SimpleGraph.Dart.fintype {V : Type u_1} {G : SimpleGraph V} [Fintype V] [DecidableRel G.Adj] :
    Fintype G.Dart
    Equations
    • One or more equations did not get rendered due to their size.
    source
    def SimpleGraph.Dart.edge {V : Type u_1} {G : SimpleGraph V} (d : G.Dart) :
    Sym2 V

    The edge associated to the dart.

    Equations
    Instances For
      source
      @[simp]
      theorem SimpleGraph.Dart.edge_mk {V : Type u_1} {G : SimpleGraph V} {p : V × V} (h : G.Adj p.1 p.2) :
      { toProd := p, adj := h }.edge = Sym2.mk p
      source
      @[simp]
      theorem SimpleGraph.Dart.edge_mem {V : Type u_1} {G : SimpleGraph V} (d : G.Dart) :
      d.edge ∈ G.edgeSet
      source
      def SimpleGraph.Dart.symm {V : Type u_1} {G : SimpleGraph V} (d : G.Dart) :
      G.Dart

      The dart with reversed orientation from a given dart.

      Equations
      • d.symm = { toProd := d.swap, adj := ⋯ }
      Instances For
        source
        @[simp]
        theorem SimpleGraph.Dart.symm_toProd {V : Type u_1} {G : SimpleGraph V} (d : G.Dart) :
        d.symm.toProd = d.swap
        source
        @[simp]
        theorem SimpleGraph.Dart.symm_mk {V : Type u_1} {G : SimpleGraph V} {p : V × V} (h : G.Adj p.1 p.2) :
        { toProd := p, adj := h }.symm = { toProd := p.swap, adj := ⋯ }
        source
        @[simp]
        theorem SimpleGraph.Dart.edge_symm {V : Type u_1} {G : SimpleGraph V} (d : G.Dart) :
        d.symm.edge = d.edge
        source
        @[simp]
        theorem SimpleGraph.Dart.edge_comp_symm {V : Type u_1} {G : SimpleGraph V} :
        SimpleGraph.Dart.edge ∘ SimpleGraph.Dart.symm = SimpleGraph.Dart.edge
        source
        @[simp]
        theorem SimpleGraph.Dart.symm_symm {V : Type u_1} {G : SimpleGraph V} (d : G.Dart) :
        d.symm.symm = d
        source
        @[simp]
        theorem SimpleGraph.Dart.symm_involutive {V : Type u_1} {G : SimpleGraph V} :
        Function.Involutive SimpleGraph.Dart.symm
        source
        theorem SimpleGraph.Dart.symm_ne {V : Type u_1} {G : SimpleGraph V} (d : G.Dart) :
        d.symm ≠ d
        source
        theorem SimpleGraph.dart_edge_eq_iff {V : Type u_1} {G : SimpleGraph V} (d₁ d₂ : G.Dart) :
        d₁.edge = d₂.edge ↔ d₁ = d₂ ∨ d₁ = d₂.symm
        source
        theorem SimpleGraph.dart_edge_eq_mk'_iff {V : Type u_1} {G : SimpleGraph V} {d : G.Dart} {p : V × V} :
        d.edge = Sym2.mk p ↔ d.toProd = p ∨ d.toProd = p.swap
        source
        theorem SimpleGraph.dart_edge_eq_mk'_iff' {V : Type u_1} {G : SimpleGraph V} {d : G.Dart} {u v : V} :
        d.edge = s(u, v) ↔ d.toProd.1 = u ∧ d.toProd.2 = v ∨ d.toProd.1 = v ∧ d.toProd.2 = u
        source
        def SimpleGraph.DartAdj {V : Type u_1} (G : SimpleGraph V) (d d' : G.Dart) :
        Prop

        Two darts are said to be adjacent if they could be consecutive darts in a walk -- that is, the first dart's second vertex is equal to the second dart's first vertex.

        Equations
        • G.DartAdj d d' = (d.toProd.2 = d'.toProd.1)
        Instances For
          source
          def SimpleGraph.dartOfNeighborSet {V : Type u_1} (G : SimpleGraph V) (v : V) (w : ↑(G.neighborSet v)) :
          G.Dart

          For a given vertex v, this is the bijective map from the neighbor set at v to the darts d with d.fst = v.

          Equations
          • G.dartOfNeighborSet v w = { toProd := (v, ↑w), adj := ⋯ }
          Instances For
            source
            @[simp]
            theorem SimpleGraph.dartOfNeighborSet_toProd {V : Type u_1} (G : SimpleGraph V) (v : V) (w : ↑(G.neighborSet v)) :
            (G.dartOfNeighborSet v w).toProd = (v, ↑w)
            source
            theorem SimpleGraph.dartOfNeighborSet_injective {V : Type u_1} (G : SimpleGraph V) (v : V) :
            Function.Injective (G.dartOfNeighborSet v)
            source
            instance SimpleGraph.nonempty_dart_top {V : Type u_1} [Nontrivial V] :
            Nonempty ⊤.Dart