Dev/MonadicDependence/WeaklySparse/Contract.lean
| 1 | import Mathlib.Combinatorics.SimpleGraph.Copy |
| 2 | import Catalog.Sparsity.Preliminaries.Contract |
| 3 | import Dev.MonadicDependence.Biclique.Contract |
| 4 | |
| 5 | namespace Dev.MonadicDependence.WeaklySparse |
| 6 | |
| 7 | open Catalog.Sparsity.Preliminaries |
| 8 | open Dev.MonadicDependence.Biclique |
| 9 | |
| 10 | /-- A graph class `π` is *weakly sparse* iff there exists a bound `k` |
| 11 | such that no graph from `π` contains the biclique of order `k` |
| 12 | (`Dev.MonadicDependence.Biclique`) as a subgraph β MΓ€hlmann Def. 13.5, |
| 13 | p. 167. |
| 14 | |
| 15 | "Contains as subgraph" is `SimpleGraph.IsContained` (Mathlib; |
| 16 | non-induced): an injective graph homomorphism `biclique k βg G`. -/ |
| 17 | def IsWeaklySparse (C : GraphClass) : Prop := |
| 18 | β k : β, β {V : Type} [DecidableEq V] [Fintype V] (G : SimpleGraph V), |
| 19 | C G β Β¬ (biclique k).IsContained G |
| 20 | |
| 21 | end Dev.MonadicDependence.WeaklySparse |
| 22 |