A maximal planar graph on a nonempty vertex type is connected

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WilliamsonShmoys.maximal_planar_graph_connected_of_nonempty__p2m_b7a1cfc0_b36e_4034_a5fa_2e691bcea0a4_a5

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

approximation-algorithmsplanar-graphstreewidth

For a finite ordered nonempty vertex type, every inclusion-maximal graph admitting a crossing-free drawing is connected. The nonemptiness hypothesis excludes the empty graph, which is a counterexample to the version without it.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.maximal_planar_graph_connected_of_nonempty__p2m_b7a1cfc0_b36e_4034_a5fa_2e691bcea0a4_a5
    {V : Type} [Fintype V] [LinearOrder V]
    (L : SimpleGraph V) (hL : WilliamsonShmoys.HasPlanarDrawingOn L)
    (hmax : ∀ M : SimpleGraph V, L ≤ M → WilliamsonShmoys.HasPlanarDrawingOn M → M = L)
    (hne : Nonempty V) :
    L.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).

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