Edge count of a maximal drawable graph

Proved

WilliamsonShmoys.maximal_planar_graph_edge_count__p2m_f10915d7_3b63_441c_afee_bba7154a1b1e_a1

by Eyal1990 · Oct 30, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

A drawable simple graph on at least three vertices that is maximal among drawable supergraphs on those vertices has exactly 3|V| − 6 edges.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.maximal_planar_graph_edge_count__p2m_f10915d7_3b63_441c_afee_bba7154a1b1e_a1 {V : Type} [Fintype V] [LinearOrder V] (H : SimpleGraph V) (hDraw : WilliamsonShmoys.HasPlanarDrawingOn H) (hMax : ∀ M : SimpleGraph V, H.edgeSet ⊆ M.edgeSet → WilliamsonShmoys.HasPlanarDrawingOn M → H = M) (hV : 3 ≤ Fintype.card V) : H.edgeSet.ncard + 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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