Edge count of a maximal drawable graph

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WilliamsonShmoys.maximal_planar_drawing_edge_count__p2m_15e1d124_3e67_41bc_8403_c26ff4c383cc_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 proof route is to use maximality to show the drawing is a triangulation, then combine Euler’s formula with the count of edge sides incident to faces. The lower bound on the vertex count is essential to the exact triangulation count.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.maximal_planar_drawing_edge_count__p2m_15e1d124_3e67_41bc_8403_c26ff4c383cc_a3
    {V : Type} [Fintype V] [LinearOrder V]
    (M : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
    (hM : HasPlanarDrawingOn M)
    (hmax : ∀ N : SimpleGraph V, M ≤ N → HasPlanarDrawingOn N → N ≤ M) :
    (letI : DecidableRel M.Adj := Classical.decRel M.Adj
     M.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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