Lax1.ExponentialSeparation
Twin-width can be exponential in treewidth
concepts/Lax1/ExponentialSeparation.lean · Lax1
Theorem
For every natural number k, there is a finite simple graph G with treewidth at most 2k + 4 and twin-width greater than 2ᵏ. Treewidth and twin-width are the parameters defined in the two prerequisite concepts.
Lean source view on GitHub
| 1 | import Lax1.Treewidth |
| 2 | import Lax1.TwinWidth |
| 3 | |
| 4 | /-! |
| 5 | --- |
| 6 | title: Twin-width can be exponential in treewidth |
| 7 | type: theorem |
| 8 | --- |
| 9 | For every natural number *k*, there is a finite simple graph *G* with |
| 10 | treewidth at most 2*k* + 4 and twin-width greater than 2ᵏ. Treewidth and |
| 11 | twin-width are the parameters defined in the two prerequisite concepts. |
| 12 | |
| 13 | # Formalization notes |
| 14 | |
| 15 | The graph is quantified over the canonical finite vertex types `Fin n`, |
| 16 | which loses no generality: every finite simple graph is isomorphic to a |
| 17 | graph on some `Fin n`. |
| 18 | -/ |
| 19 | |
| 20 | namespace Lax1.ExponentialSeparation |
| 21 | |
| 22 | /-- For every `k`, some finite graph has treewidth at most `2 * k + 4` and |
| 23 | twin-width greater than `2 ^ k`. -/ |
| 24 | axiom twin_width_can_be_exponential_in_treewidth (k : ℕ) : |
| 25 | ∃ n : ℕ, ∃ G : SimpleGraph (Fin n), |
| 26 | Lax1.Treewidth.treewidth G ≤ 2 * k + 4 ∧ |
| 27 | 2 ^ k < Lax1.TwinWidth.twinWidth G |
| 28 | |
| 29 | end Lax1.ExponentialSeparation |
| 30 |
Formalization notes
The graph is quantified over the canonical finite vertex types Fin n,
which loses no generality: every finite simple graph is isomorphic to a
graph on some Fin n.
Imported
Imported by
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Mathlib imports
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