| import OAI.Combinatorics.HadwigerMatching.Definitions |
| import OAI.Combinatorics.HadwigerMatching.BranchCounting |
|
|
| namespace OAI |
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| /-! Cliques and touching matchings formed by singleton and doubleton branches. -/ |
| namespace HadwigerMatching |
| open scoped BigOperators |
| variable {V : Type*} [DecidableEq V] {t : ℕ} |
|
|
| def singletonVertices (B : Fin t → Finset V) : Finset V := |
| (Finset.univ.filter fun i => (B i).card = 1).biUnion B |
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| def twoVertexBranches (B : Fin t → Finset V) : Finset (Finset V) := |
| (Finset.univ.filter fun i => (B i).card = 2).image B |
|
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| theorem mem_singletonVertices (B : Fin t → Finset V) (v : V) : |
| v ∈ singletonVertices B ↔ ∃ i, (B i).card = 1 ∧ v ∈ B i := by |
| simp [singletonVertices] |
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|
| theorem mem_twoVertexBranches (B : Fin t → Finset V) (e : Finset V) : |
| e ∈ twoVertexBranches B ↔ ∃ i, (B i).card = 2 ∧ B i = e := by |
| simp [twoVertexBranches] |
|
|
| omit [DecidableEq V] in |
| /-- Nonemptiness is essential: distinct empty branches would not be injective. -/ |
| theorem branches_injective (B : Fin t → Finset V) |
| (hne : ∀ i, (B i).Nonempty) |
| (hdisj : Pairwise (fun i j => Disjoint (B i) (B j))) : |
| Function.Injective B := by |
| intro i j he |
| by_contra hij |
| obtain ⟨v, hv⟩ := hne i |
| exact Finset.disjoint_left.mp (hdisj hij) hv (he ▸ hv) |
|
|
| theorem card_singletonVertices (B : Fin t → Finset V) |
| (hdisj : Pairwise (fun i j => Disjoint (B i) (B j))) : |
| (singletonVertices B).card = singletonBranchCount B := by |
| unfold singletonVertices singletonBranchCount |
| rw [Finset.card_biUnion (fun i _ j _ hij => hdisj hij)] |
| calc |
| (∑ i ∈ Finset.univ.filter (fun i => (B i).card = 1), (B i).card) = |
| ∑ i ∈ Finset.univ.filter (fun i => (B i).card = 1), 1 := by |
| apply Finset.sum_congr rfl |
| intro i hi |
| exact (Finset.mem_filter.mp hi).2 |
| _ = _ := by simp |
|
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| theorem card_twoVertexBranches (B : Fin t → Finset V) |
| (hne : ∀ i, (B i).Nonempty) |
| (hdisj : Pairwise (fun i j => Disjoint (B i) (B j))) : |
| (twoVertexBranches B).card = doubletonBranchCount B := by |
| unfold twoVertexBranches doubletonBranchCount |
| exact Finset.card_image_of_injective _ (branches_injective B hne hdisj) |
|
|
| omit [DecidableEq V] in |
| theorem singleton_branch_eq (B : Fin t → Finset V) {i : Fin t} {v : V} |
| (hi : (B i).card = 1) (hv : v ∈ B i) : B i = {v} := by |
| obtain ⟨w, hw⟩ := Finset.card_eq_one.mp hi |
| have hvw : v = w := by simpa [hw] using hv |
| simpa [hvw] using hw |
|
|
| theorem singletonVertices_isClique (G : SimpleGraph V) (B : Fin t → Finset V) |
| (htouch : ∀ i j, i ≠ j → ∃ v ∈ B i, ∃ w ∈ B j, G.Adj v w) : |
| G.IsClique (singletonVertices B : Set V) := by |
| intro v hv w hw hvw |
| obtain ⟨i, hi, hvi⟩ := (mem_singletonVertices B v).mp hv |
| obtain ⟨j, hj, hwj⟩ := (mem_singletonVertices B w).mp hw |
| have hBi := singleton_branch_eq B hi hvi |
| have hBj := singleton_branch_eq B hj hwj |
| have hij : i ≠ j := by |
| intro he |
| have hmem : v ∈ B j := he ▸ hvi |
| have hvw' : v = w := by simpa [hBj] using hmem |
| exact hvw hvw' |
| obtain ⟨x, hx, y, hy, hxy⟩ := htouch i j hij |
| have hxv : x = v := by simpa [hBi] using hx |
| have hyw : y = w := by simpa [hBj] using hy |
| simpa [hxv, hyw] using hxy |
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|
| theorem twoVertexBranches_isTouchingMatching (G : SimpleGraph V) |
| (B : Fin t → Finset V) |
| (hdisj : Pairwise (fun i j => Disjoint (B i) (B j))) |
| (htouch : ∀ i j, i ≠ j → ∃ v ∈ B i, ∃ w ∈ B j, G.Adj v w) |
| (hclique : ∀ i, (B i).card = 2 → G.IsClique (B i : Set V)) : |
| IsTouchingMatching G (twoVertexBranches B) := by |
| refine ⟨?_, ?_, ?_⟩ |
| · intro e he |
| obtain ⟨i, hi, rfl⟩ := (mem_twoVertexBranches B e).mp he |
| exact ⟨hi, hclique i hi⟩ |
| · intro e he f hf hef |
| obtain ⟨i, _, rfl⟩ := (mem_twoVertexBranches B e).mp he |
| obtain ⟨j, _, rfl⟩ := (mem_twoVertexBranches B f).mp hf |
| exact hdisj (fun hij => hef (congrArg B hij)) |
| · intro e he f hf hef |
| obtain ⟨i, _, rfl⟩ := (mem_twoVertexBranches B e).mp he |
| obtain ⟨j, _, rfl⟩ := (mem_twoVertexBranches B f).mp hf |
| exact htouch i j (fun hij => hef (congrArg B hij)) |
|
|
| theorem twoVertexBranches_disjoint_singletonVertices (B : Fin t → Finset V) |
| (hdisj : Pairwise (fun i j => Disjoint (B i) (B j))) |
| {e : Finset V} (he : e ∈ twoVertexBranches B) : |
| Disjoint e (singletonVertices B) := by |
| obtain ⟨i, hi, rfl⟩ := (mem_twoVertexBranches B e).mp he |
| apply Finset.disjoint_left.mpr |
| intro v hvi hv |
| obtain ⟨j, hj, hvj⟩ := (mem_singletonVertices B v).mp hv |
| have hij : i ≠ j := by |
| intro heq |
| have : (2 : ℕ) = 1 := hi.symm.trans ((congrArg (fun k => (B k).card) heq).trans hj) |
| exact (by decide : (2 : ℕ) ≠ 1) this |
| exact Finset.disjoint_left.mp (hdisj hij) hvi hvj |
|
|
| theorem singletonVertices_touch_twoVertexBranches (G : SimpleGraph V) |
| (B : Fin t → Finset V) |
| (htouch : ∀ i j, i ≠ j → ∃ v ∈ B i, ∃ w ∈ B j, G.Adj v w) |
| {v : V} (hv : v ∈ singletonVertices B) |
| {e : Finset V} (he : e ∈ twoVertexBranches B) : |
| ∃ w ∈ e, G.Adj v w := by |
| obtain ⟨i, hi, hvi⟩ := (mem_singletonVertices B v).mp hv |
| obtain ⟨j, hj, rfl⟩ := (mem_twoVertexBranches B e).mp he |
| have hij : i ≠ j := by |
| intro heq |
| have : (1 : ℕ) = 2 := hi.symm.trans ((congrArg (fun k => (B k).card) heq).trans hj) |
| exact (by decide : (1 : ℕ) ≠ 2) this |
| obtain ⟨x, hx, w, hw, hxw⟩ := htouch i j hij |
| have hxv : x = v := by simpa [singleton_branch_eq B hi hvi] using hx |
| exact ⟨w, hw, hxv ▸ hxw⟩ |
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|
| end HadwigerMatching |
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| end OAI |
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