Edge count of a maximal drawable graph

Proved

WilliamsonShmoys.maximal_drawable_graph_edge_count__p2m_f0debea5_402c_48d1_9751_ec911f78abab_a1

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

approximation-algorithmsplanar-graphstreewidth

A drawable graph on a finite ordered vertex type with at least three vertices, maximal among drawable graphs on that vertex type, has exactly three times the number of vertices minus six edges.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.maximal_drawable_graph_edge_count__p2m_f0debea5_402c_48d1_9751_ec911f78abab_a1 {V : Type} [Fintype V] [LinearOrder V] (N : SimpleGraph V) [DecidableRel N.Adj] (hcard : 3 ≤ Fintype.card V) (hN : HasPlanarDrawingOn N) (hmax : ∀ (P : SimpleGraph V) [DecidableRel P.Adj], N ≤ P → HasPlanarDrawingOn P → P = N) : N.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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