Lower edge bound for an inclusion-maximal drawable graph

Proved

maximal_drawable_graph_edge_lower_bound__p2m_d1879139_c8f2_40b8_b730_a1c2d5f0f3db_a2

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

approximation-algorithmsplanar-graphstreewidth

An inclusion-maximal drawable simple graph on at least three finite, linearly ordered vertices has at least three times the number of vertices minus six edges.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

Formal statement

theorem maximal_drawable_graph_edge_lower_bound__p2m_d1879139_c8f2_40b8_b730_a1c2d5f0f3db_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (M : SimpleGraph V)
    (hM : WilliamsonShmoys.HasPlanarDrawingOn M)
    (hmax : ∀ J : SimpleGraph V,
      WilliamsonShmoys.HasPlanarDrawingOn J → M ≤ J → J ≤ M)
    (hn : 3 ≤ Fintype.card V) :
    (letI : DecidableRel M.Adj := Classical.decRel M.Adj
     3 * Fintype.card V - 6 ≤ M.edgeFinset.card) := 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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