| 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 |
|
|
