Maximal drawable graph is connected

Proved

WilliamsonShmoys.maximal_drawable_connected__p2m_2752f1ab_5752_4af7_9412_6b19a571a389_a1

by Eyal1990 · Nov 1, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

A drawable graph on at least three vertices that is maximal among drawable supergraphs on the same vertex type is connected.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib
open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.maximal_drawable_connected__p2m_2752f1ab_5752_4af7_9412_6b19a571a389_a1 {V : Type} [Fintype V] [LinearOrder V] (M : SimpleGraph V) (dM : DecidableRel M.Adj) (hM : HasPlanarDrawingOn M) (hcard : 3 <= Fintype.card V) (hmax : forall (N : SimpleGraph V) (dN : DecidableRel N.Adj), M <= N -> HasPlanarDrawingOn N -> N = M) : M.Connected := by sorry

Source

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).

View graph