File size: 2,399 Bytes
e82fac3 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 | import Mathlib.Combinatorics.SimpleGraph.Matching
import Lean.Elab.Tactic.Omega
namespace OAI
/-! Perfect-matching-preserving port-path degree reduction for finite simple graphs. -/
noncomputable section
namespace Matching205
universe u
variable {V : Type u} [Fintype V] (G : SimpleGraph V) [DecidableRel G.Adj]
abbrev arity (v : V) : ℕ := Fintype.card (G.neighborSet v)
abbrev Vertex := (v : V) × Fin (2 * arity G v - 1)
def port (v : V) (w : G.neighborSet v) : Vertex G :=
⟨v, ⟨2 * ((Fintype.equivFin (G.neighborSet v)) w).val, by
have h := ((Fintype.equivFin (G.neighborSet v)) w).isLt
dsimp [arity] at *
omega⟩⟩
def Internal (x y : Vertex G) : Prop :=
x.1 = y.1 ∧ (x.2.val + 1 = y.2.val ∨ y.2.val + 1 = x.2.val)
def External (x y : Vertex G) : Prop :=
∃ h : G.Adj x.1 y.1,
x = port G x.1 ⟨y.1, h⟩ ∧ y = port G y.1 ⟨x.1, h.symm⟩
lemma internal_symm {x y : Vertex G} (h : Internal G x y) : Internal G y x :=
⟨h.1.symm, h.2.symm⟩
lemma external_symm {x y : Vertex G} (h : External G x y) : External G y x := by
obtain ⟨h, hx, hy⟩ := h
exact ⟨h.symm, hy, hx⟩
def graph : SimpleGraph (Vertex G) where
Adj x y := Internal G x y ∨ External G x y
symm := ⟨fun _ _ h => h.elim (fun h => Or.inl (internal_symm G h))
(fun h => Or.inr (external_symm G h))⟩
loopless := ⟨by
intro x h
rcases h with h | ⟨h, _, _⟩
· rcases h.2 with h | h <;> omega
· exact G.irrefl h⟩
lemma port_adj {v w : V} (h : G.Adj v w) :
(graph G).Adj (port G v ⟨w, h⟩) (port G w ⟨v, h.symm⟩) :=
Or.inr ⟨h, rfl, rfl⟩
def decode (M : (graph G).Subgraph) : G.Subgraph where
verts := Set.univ
Adj v w := ∃ h : G.Adj v w,
M.Adj (port G v ⟨w, h⟩) (port G w ⟨v, h.symm⟩)
adj_sub := fun ⟨h, _⟩ => h
edge_vert := fun _ => Set.mem_univ _
symm := ⟨by
intro v w h
obtain ⟨h, hm⟩ := h
exact ⟨h.symm, hm.symm⟩⟩
lemma decode_adj_iff (M : (graph G).Subgraph) (v w : V) :
(decode G M).Adj v w ↔ ∃ h : G.Adj v w,
M.Adj (port G v ⟨w, h⟩) (port G w ⟨v, h.symm⟩) := Iff.rfl
lemma card_fiber (v : V) : Fintype.card (Fin (2 * arity G v - 1)) =
2 * G.degree v - 1 := by
rw [Fintype.card_fin]
change 2 * Fintype.card (G.neighborSet v) - 1 = 2 * G.degree v - 1
rw [SimpleGraph.card_neighborSet_eq_degree]
end Matching205
end
end OAI
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