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Distance to complete graphs

Lax825442.DistanceToClique · concepts/Lax825442/DistanceToClique.lean · lax-825442

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    Definition

    The distance to complete graphs is the minimum number of vertices whose deletion leaves a complete induced graph. Completeness uses mathlib's top graph, which has every edge between distinct vertices. The empty graph is complete under this convention, so deleting all vertices always suffices.

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    1import Mathlib.Combinatorics.SimpleGraph.Finite
    2import Mathlib.Order.Lattice.Nat
    3
    4/-!
    5---
    6title: Distance to complete graphs
    7type: definition
    8---
    9The distance to complete graphs is the minimum number of vertices whose
    10deletion leaves a complete induced graph. Completeness uses mathlib's top
    11graph, which has every edge between distinct vertices. The empty graph is
    12complete under this convention, so deleting all vertices always suffices.
    13-/
    14
    15namespace Lax825442.DistanceToClique
    16
    17/-- Minimum number of vertices whose deletion leaves a complete graph. -/
    18noncomputable def distanceToClique {V : Type} [Fintype V] [DecidableEq V]
    19 (G : SimpleGraph V) : ℕ :=
    20 sInf {k : ℕ | ∃ S : Finset V, (S.card = k) ∧
    21 (G.induce {v | v ∉ S} = ⊤)}
    22
    23end Lax825442.DistanceToClique
    24
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