File size: 3,139 Bytes
e320e82 | 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 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 | import OAI.Combinatorics.ListHadwiger.Basic
namespace OAI
/-! Clique geometry of singleton and doubleton branches and finite clique-minor models. -/
namespace HadwigerMatching
variable {V : Type} {G : SimpleGraph V}
/-- 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⟩
end HadwigerMatching
end OAI
|