A drawable extension with maximum edge count

Proved

WilliamsonShmoys.exists_maximal_planar_supergraph_by_edge_card__p2m_9ce78458_17c3_4c03_93dc_57f21947e1ab_a2

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

approximation-algorithmsplanar-graphstreewidth

Among drawable supergraphs of a given drawable graph on a finite ordered vertex type, one has at least as many edges as every other drawable supergraph.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.exists_maximal_planar_supergraph_by_edge_card__p2m_9ce78458_17c3_4c03_93dc_57f21947e1ab_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V) (hdraw : WilliamsonShmoys.HasPlanarDrawingOn G) :
    ∃ H : SimpleGraph V, G ≤ H ∧ WilliamsonShmoys.HasPlanarDrawingOn H ∧
      ∀ K : SimpleGraph V, G ≤ K → WilliamsonShmoys.HasPlanarDrawingOn K →
        Set.ncard H.edgeSet ≥ Set.ncard K.edgeSet := 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