Existence of an inclusion-maximal drawable supergraph
ProvedWilliamsonShmoys.maximal_planar_supergraph_exists__p2m_3461d3ef_e829_4302_a17d_0f8fc9f6195e_a1approximation-algorithmsplanar-graphstreewidth
Every drawable graph on a finite vertex type has a drawable supergraph on the same vertices that is inclusion-maximal among drawable supergraphs.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.maximal_planar_supergraph_exists__p2m_3461d3ef_e829_4302_a17d_0f8fc9f6195e_a1
{V : Type} [Fintype V] [LinearOrder V]
(H : SimpleGraph V) (hH : HasPlanarDrawingOn H)
(hcard : 3 <= Fintype.card V) :
exists J : SimpleGraph V,
And (HasPlanarDrawingOn J)
(And (H <= J)
(forall K : SimpleGraph V,
HasPlanarDrawingOn K -> J <= K -> K <= J)) := 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).