A drawable graph has a maximal drawable extension

Proved

WilliamsonShmoys.drawable_graph_has_maximal_extension__p2m_637d8540_4fa2_4bdf_8eaf_997caf9c33c1_a2

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

approximation-algorithmsplanar-graphstreewidth

Every drawable finite graph has a drawable supergraph on the same vertex type that is maximal under graph inclusion among drawable graphs.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib
open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.drawable_graph_has_maximal_extension__p2m_637d8540_4fa2_4bdf_8eaf_997caf9c33c1_a2 {V : Type} [Fintype V] [LinearOrder V] (N : SimpleGraph V) [DecidableRel N.Adj] (hN : HasPlanarDrawingOn N) : ∃ (P : SimpleGraph V), N ≤ P ∧ HasPlanarDrawingOn P ∧ (∀ Q : SimpleGraph V, P ≤ Q → HasPlanarDrawingOn Q → Q = P) := 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