Maximal planar edge lower bound

Proved

WilliamsonShmoys.maximal_planar_edge_lower_bound__p2m_3ea6bb1c_a422_4d1c_b17f_ef9a0e48b692_a1

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

approximation-algorithmsplanar-graphstreewidth

A drawable graph on at least three vertices that is maximal among drawable supergraphs on the same vertex type has at least 3|V|−6 edges. Extend its drawing to a triangulation on the same vertices; maximality forces the extension to have no additional edges, and Euler’s formula gives the bound.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.maximal_planar_edge_lower_bound__p2m_3ea6bb1c_a422_4d1c_b17f_ef9a0e48b692_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) (hK : HasPlanarDrawingOn K)
    (hmax : ∀ L : SimpleGraph V, K ≤ L → HasPlanarDrawingOn L → L ≤ K)
    (hV : 3 ≤ Fintype.card V) :
    letI : DecidableRel K.Adj := Classical.decRel K.Adj
    letI : Fintype ↑K.edgeSet := Fintype.ofFinite _
    3 * Fintype.card V ≤ K.edgeFinset.card + 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).

View graph