A maximal drawable extension is connected
DisprovedWilliamsonShmoys.maximal_planar_extension_connected__p2m_986cca2a_af96_41ca_97f1_9fc97c42977c_a2approximation-algorithmsplanar-graphstreewidth
If H is drawable and maximal among drawable supergraphs of K on the same finite vertex type, then H is connected.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib
Formal statement
theorem WilliamsonShmoys.maximal_planar_extension_connected__p2m_986cca2a_af96_41ca_97f1_9fc97c42977c_a2
{V : Type} [Fintype V] [LinearOrder V]
(K H : SimpleGraph V)
(hH : WilliamsonShmoys.HasPlanarDrawingOn H)
(hKH : K ≤ H)
(hmax : ∀ J : SimpleGraph V,
K ≤ J →
WilliamsonShmoys.HasPlanarDrawingOn J →
H ≤ J → J = H) :
H.Connected := 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).