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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 | 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
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