Subgraphs of maximal drawable graphs obey the planar edge bound

Proved

WilliamsonShmoys.subgraph_of_maximal_planar_drawing_obeys_edge_bound__p2m_40c1c52c_c498_4287_a557_1efecd0d6ca7_a3

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

approximation-algorithmsplanar-graphstreewidth

If a drawable graph on at least three vertices is maximal among drawable graphs on those vertices, every subgraph has at most three times the vertex count minus six edges.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.subgraph_of_maximal_planar_drawing_obeys_edge_bound__p2m_40c1c52c_c498_4287_a557_1efecd0d6ca7_a3
    {V : Type} [Fintype V] [LinearOrder V]
    (G H : SimpleGraph V) [DecidableRel G.Adj] [DecidableRel H.Adj]
    (hcard : 3 ≤ Fintype.card V)
    (hsub : G ≤ H)
    (hH : WilliamsonShmoys.HasPlanarDrawingOn H)
    (hmax : ∀ K : SimpleGraph V, H ≤ K →
      WilliamsonShmoys.HasPlanarDrawingOn K → K = H) :
    G.edgeFinset.card + 6 ≤ 3 * Fintype.card V := 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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