Existence of a maximum drawable supergraph
ProvedWilliamsonShmoys.exists_maximum_drawable_supergraph__p2m_14ab409d_ba46_4520_b2b1_fae32ccc2155_a1approximation-algorithmsplanar-graphstreewidth
Every drawable finite simple graph has a drawable supergraph on the same vertices whose edge count is maximum among its drawable supergraphs.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.exists_maximum_drawable_supergraph__p2m_14ab409d_ba46_4520_b2b1_fae32ccc2155_a1
{V : Type} [Fintype V] [LinearOrder V]
(M : SimpleGraph V)
(hM : WilliamsonShmoys.HasPlanarDrawingOn M) :
∃ N : SimpleGraph V,
M ≤ N ∧ WilliamsonShmoys.HasPlanarDrawingOn N ∧
∀ K : SimpleGraph V, M ≤ K → WilliamsonShmoys.HasPlanarDrawingOn K →
K.edgeSet.ncard ≤ N.edgeSet.ncard := 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).