Finite maximum among drawable supergraphs
ProvedWilliamsonShmoys.exists_max_drawable_extension_by_arc_count__p2m_c50204fa_335d_4575_bac2_23cf83d1b5e4_a1approximation-algorithmsplanar-graphstreewidth
Among all drawable supergraphs of a drawable finite graph, one maximizes the number of directed adjacent vertex pairs.
Preamble
import Mathlib import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.exists_max_drawable_extension_by_arc_count__p2m_c50204fa_335d_4575_bac2_23cf83d1b5e4_a1
{V : Type} [Fintype V] [LinearOrder V]
(N : SimpleGraph V) (hNdraw : HasPlanarDrawingOn N) :
∃ K : SimpleGraph V,
N ≤ K ∧ HasPlanarDrawingOn K ∧
∀ L : SimpleGraph V, N ≤ L → HasPlanarDrawingOn L →
Nat.card {p : V × V // K.Adj p.1 p.2} ≤
Nat.card {p : V × V // L.Adj p.1 p.2} := by sorrySource
David P. Williamson and David B. Shmoys, The Design of Approximation Algorithms, Cambridge University Press, 2011, author electronic manuscript, Section 10.2, proof of Theorem 10.11, pp. 270-272. https://doi.org/10.1017/CBO9780511921735. Bounded tree-width of the pieces: Robertson–Seymour (planar graphs of radius r have tree-width at most 3r+1) / Bodlaender (k-outerplanar graphs have tree-width at most 3k-1).