Existence of an inclusion-maximal drawable extension
ProvedWilliamsonShmoys.exists_maximal_planar_extension__p2m_71345845_0812_4936_9e60_f9671145c018_a2approximation-algorithmsplanar-graphstreewidth
Every drawable simple graph on a finite linearly ordered vertex type has a drawable supergraph on the same vertices that is maximal under graph inclusion among drawable graphs. A proof route is to maximize edge count over the finite set of drawable supergraphs of the input graph. This step applies even when the input graph is empty or disconnected.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.exists_maximal_planar_extension__p2m_71345845_0812_4936_9e60_f9671145c018_a2
{V : Type} [Fintype V] [LinearOrder V]
(L : SimpleGraph V) (hL : HasPlanarDrawingOn L) :
∃ T : SimpleGraph V, L ≤ T ∧ HasPlanarDrawingOn T ∧
∀ U : SimpleGraph V, T ≤ U → HasPlanarDrawingOn U → U = T := 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).