Edge lower bound for a maximal drawable graph

Proved

maximal_drawable_graph_edge_lower_bound__p2m_d2caba73_4b7e_439a_8f09_ce231377a6dc_a2

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

approximation-algorithmsplanar-graphstreewidth

A drawable graph on at least three vertices that is maximal among drawable supergraphs has at least 3|V|−6 edges. Prove this by deriving triangulation and the needed connectivity from maximality, then applying planar counting.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

Formal statement

theorem maximal_drawable_graph_edge_lower_bound__p2m_d2caba73_4b7e_439a_8f09_ce231377a6dc_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) (dK : DecidableRel K.Adj)
    (hK : WilliamsonShmoys.HasPlanarDrawingOn K)
    (hn : 3 <= Fintype.card V)
    (hmax : ∀ L : SimpleGraph V,
      WilliamsonShmoys.HasPlanarDrawingOn L → K ≤ L → L ≤ K) :
    letI : DecidableRel K.Adj := dK
    3 * Fintype.card V - 6 <= K.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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