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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 | import Mathlib.Combinatorics.SimpleGraph.Clique
namespace OAI
namespace HadwigerMatching
universe u
variable {V : Type u}
/-- A finite matching whose edges are pairwise touching: each member is a
two-vertex clique, and distinct members are disjoint and joined by an edge. -/
def IsTouchingMatching (G : SimpleGraph V) (M : Finset (Finset V)) : Prop :=
(∀ e ∈ M, e.card = 2 ∧ G.IsClique (e : Set V)) ∧
(∀ e ∈ M, ∀ f ∈ M, e ≠ f → Disjoint e f) ∧
(∀ e ∈ M, ∀ f ∈ M, e ≠ f →
∃ v ∈ e, ∃ w ∈ f, G.Adj v w)
/-- The supremum of the cardinalities of pairwise touching matchings.
For finite vertex types, this supremum is attained. -/
noncomputable def connectedMatchingNumber (G : SimpleGraph V) : ℕ :=
sSup {n : ℕ | ∃ M : Finset (Finset V), IsTouchingMatching G M ∧ M.card = n}
end HadwigerMatching
end OAI
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