An edge-maximal drawable graph has the triangulation edge count

Proved

WilliamsonShmoys.maximal_planar_edge_count__p2m_dcc51884_0da8_465d_ba77_04c61fbeb5d3_a1

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

approximation-algorithmsplanar-graphstreewidth

If a drawable simple graph on at least three vertices has no strictly larger drawable supergraph on the same vertex type, then its edge count is exactly three times the number of vertices minus six.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.maximal_planar_edge_count__p2m_dcc51884_0da8_465d_ba77_04c61fbeb5d3_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (U : SimpleGraph V) (hV : 3 ≤ Fintype.card V)
    (hU : HasPlanarDrawingOn U)
    (hmax : ∀ H : SimpleGraph V, U ≤ H → HasPlanarDrawingOn H → H ≤ U) :
    letI : DecidableRel U.Adj := Classical.decRel U.Adj
    U.edgeFinset.card = 3 * Fintype.card V - 6 := 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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