Choose an edge-maximal drawable supergraph

Proved

WilliamsonShmoys.planarDrawing_edge_maximal_supergraph__p2m_dddaee3a_0582_44d9_9566_a5dd1d115710_a2

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

approximation-algorithmsplanar-graphstreewidth

Every drawable graph on a finite vertex type has a drawable supergraph that is maximal among drawable supergraphs under edge inclusion.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planarDrawing_edge_maximal_supergraph__p2m_dddaee3a_0582_44d9_9566_a5dd1d115710_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V)
    (hK : WilliamsonShmoys.HasPlanarDrawingOn K) :
    ∃ M : SimpleGraph V,
      K ≤ M ∧ WilliamsonShmoys.HasPlanarDrawingOn M ∧
        ∀ H : SimpleGraph V,
          M ≤ H → WilliamsonShmoys.HasPlanarDrawingOn H → H ≤ M := 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