| import OAI.Combinatorics.PortMatching.Extension |
| import OAI.Combinatorics.PortMatching.Decoding |
| import OAI.Combinatorics.PortMatching.Size |
|
|
| 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] |
|
|
| theorem perfectMatching_forward (M : G.Subgraph) (hM : M.IsPerfectMatching) : |
| ∃ N : (graph G).Subgraph, N.IsPerfectMatching := |
| ⟨Forward.extend G M hM, Forward.extend_perfect G M hM⟩ |
|
|
| theorem perfectMatching_reverse (hpos : ∀ v : V, 0 < G.degree v) |
| (N : (graph G).Subgraph) (hN : N.IsPerfectMatching) : |
| ∃ M : G.Subgraph, M.IsPerfectMatching := by |
| have ha : ∀ v : V, 0 < arity G v := by |
| intro v |
| simpa only [arity, SimpleGraph.card_neighborSet_eq_degree] using hpos v |
| exact ⟨decode G N, decode_isPerfectMatching G ha hN⟩ |
|
|
| theorem perfectMatching_iff (hpos : ∀ v : V, 0 < G.degree v) : |
| (∃ M : G.Subgraph, M.IsPerfectMatching) ↔ |
| ∃ N : (graph G).Subgraph, N.IsPerfectMatching := by |
| constructor |
| · rintro ⟨M, hM⟩ |
| exact perfectMatching_forward G M hM |
| · rintro ⟨N, hN⟩ |
| exact perfectMatching_reverse G hpos N hN |
|
|
| theorem simpleGraphReduction (hpos : ∀ v : V, 0 < G.degree v) : |
| ((∃ M : G.Subgraph, M.IsPerfectMatching) ↔ |
| ∃ N : (graph G).Subgraph, N.IsPerfectMatching) ∧ |
| (∀ x : Vertex G, (graph G).degree x ≤ 3) ∧ |
| Fintype.card (Vertex G) = 4 * G.edgeFinset.card - Fintype.card V := |
| ⟨perfectMatching_iff G hpos, graph_degree_le_three G, card_vertex G hpos⟩ |
|
|
| end Matching205 |
|
|
| end |
|
|
| end OAI |
|
|
