An inclusion-maximal planar drawing has the triangulation edge count

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WilliamsonShmoys.maximal_planar_drawing_has_triangulation_edge_count__p2m_64226b9d_e479_429e_9ec8_cb1b67946ff3_a3

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

approximation-algorithmsplanar-graphstreewidth

An inclusion-maximal drawable simple graph on at least three vertices has exactly 3|V| − 6 edges. The face-triangulation and Euler-counting argument is the core geometric work remaining after the parent’s maximal extension step.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.maximal_planar_drawing_has_triangulation_edge_count__p2m_64226b9d_e479_429e_9ec8_cb1b67946ff3_a3
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V) (dG : DecidableRel G.Adj)
    (hG : HasPlanarDrawingOn G)
    (hn : 3 ≤ Fintype.card V)
    (hmax : ∀ (K : SimpleGraph V) (dK : DecidableRel K.Adj),
      letI : DecidableRel G.Adj := dG
      letI : DecidableRel K.Adj := dK
      HasPlanarDrawingOn K →
      G.edgeFinset ⊆ K.edgeFinset →
      K.edgeFinset = G.edgeFinset) :
    letI : DecidableRel G.Adj := dG
    G.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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