| import OAI.Combinatorics.ListHadwiger.Basic |
|
|
| namespace OAI |
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| /-! Clique geometry of singleton and doubleton branches and finite clique-minor models. -/ |
| namespace HadwigerMatching |
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|
| variable {V : Type} {G : SimpleGraph V} |
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|
| /-- A connected induced graph on precisely two vertices contains their edge. -/ |
| theorem connected_card_two_isClique [DecidableEq V] (S : Finset V) |
| (hS : S.card = 2) (hc : (G.induce (S : Set V)).Connected) : |
| G.IsClique (S : Set V) := by |
| obtain ⟨a, b, hab, rfl⟩ := Finset.card_eq_two.mp hS |
| have hedge : G.Adj a b := by |
| let a' : ({a, b} : Finset V) := ⟨a, by simp⟩ |
| let b' : ({a, b} : Finset V) := ⟨b, by simp⟩ |
| have hne : a' ≠ b' := by |
| intro h |
| exact hab (congrArg Subtype.val h) |
| obtain ⟨p⟩ := hc.preconnected a' b' |
| have hadj := p.adj_snd (p.not_nil_of_ne hne) |
| have hsnd : (p.snd : V) = a ∨ (p.snd : V) = b := by |
| simpa using p.snd.property |
| rcases hsnd with hsnd | hsnd |
| · change G.Adj a (p.snd : V) at hadj |
| rw [hsnd] at hadj |
| exact (G.irrefl hadj).elim |
| · simpa [a', hsnd] using hadj |
| intro x hx y hy hxy |
| have hx' : x = a ∨ x = b := by simpa using hx |
| have hy' : y = a ∨ y = b := by simpa using hy |
| rcases hx' with rfl | rfl <;> rcases hy' with rfl | rfl |
| · exact (hxy rfl).elim |
| · exact hedge |
| · exact hedge.symm |
| · exact (hxy rfl).elim |
|
|
| /-- The vertices of singleton branches form a clique in `G`. -/ |
| theorem singleton_vertices_isClique {ι : Type*} (B : ι → Set V) |
| (htouch : ∀ i j, i ≠ j → ∃ v ∈ B i, ∃ w ∈ B j, G.Adj v w) : |
| G.IsClique {v | ∃ i, B i = {v}} := by |
| rintro x ⟨i, hi⟩ y ⟨j, hj⟩ hxy |
| have hij : i ≠ j := by |
| intro h |
| subst j |
| have : x ∈ ({y} : Set V) := by rw [← hj, hi]; simp |
| exact hxy (Set.mem_singleton_iff.mp this) |
| obtain ⟨v, hv, w, hw, hvw⟩ := htouch i j hij |
| have hvx : v = x := by simpa [hi] using hv |
| have hwy : w = y := by simpa [hj] using hw |
| simpa [hvx, hwy] using hvw |
|
|
| /-- On a finite vertex type, clique-minor branches can be represented by finite sets. -/ |
| theorem finite_branch_model [Fintype V] {t : ℕ} |
| (h : LinearListHadwiger.HasCliqueMinor G t) : |
| ∃ B : Fin t → Finset V, |
| (∀ i, (G.induce (B i : Set V)).Connected) ∧ |
| (∀ i, (B i).Nonempty) ∧ |
| Pairwise (fun i j => Disjoint (B i) (B j)) ∧ |
| (∀ i j, i ≠ j → ∃ v ∈ B i, ∃ w ∈ B j, G.Adj v w) := by |
| classical |
| obtain ⟨A, hconn, hdisj, htouch⟩ := h |
| let B : Fin t → Finset V := fun i => (A i).toFinset |
| have hcoe (i : Fin t) : (B i : Set V) = A i := by simp [B] |
| refine ⟨B, ?_, ?_, ?_, ?_⟩ |
| · intro i |
| rw [hcoe] |
| exact hconn i |
| · intro i |
| obtain ⟨v⟩ := (hconn i).nonempty |
| exact ⟨v.val, Set.mem_toFinset.mpr v.property⟩ |
| · intro i j hij |
| apply Finset.disjoint_left.mpr |
| intro v hvi hvj |
| exact Set.disjoint_left.mp (hdisj i j hij) |
| (by simpa [B] using hvi) (by simpa [B] using hvj) |
| · intro i j hij |
| obtain ⟨v, hv, w, hw, hvw⟩ := htouch i j hij |
| exact ⟨v, by simpa [B] using hv, w, by simpa [B] using hw, hvw⟩ |
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| end HadwigerMatching |
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| end OAI |
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