Edge count of a maximal drawn graph

Proved

WilliamsonShmoys.maximal_planar_graph_edge_count__p2m_2cd0a648_3850_4669_bbb5_b62598f0c8b2_a1

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

approximation-algorithmsplanar-graphstreewidth

A graph with the specified drawing that is maximal among drawn supergraphs on its vertex set has exactly three times its vertex count minus six edges, when the vertex count is at least three.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.maximal_planar_graph_edge_count__p2m_2cd0a648_3850_4669_bbb5_b62598f0c8b2_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
    (hplanar : HasPlanarDrawingOn H)
    (hmax : ∀ K : SimpleGraph V, H ≤ K → HasPlanarDrawingOn K → K = H) :
    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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