Connected planar simple graph edge bound

Proved

WilliamsonShmoys.connected_planar_edge_count_upper__p2m_e8818a47_c6bb_451e_9488_1f3faaa47e8f_a3

by Eyal1990 · Oct 31, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

For the connected case, derive a plane embedding from the drawing, apply Euler’s formula, and count face-boundary incidences. Simplicity and the at-least-three-vertices hypothesis give boundary length at least three for every face. The remaining core obligation is formalizing the embedding, Euler, and face-counting steps for the explicit arc-based drawing predicate.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.connected_planar_edge_count_upper__p2m_e8818a47_c6bb_451e_9488_1f3faaa47e8f_a3
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V)
    (hK : HasPlanarDrawingOn K)
    (hconn : K.Connected)
    (hcard : 3 ≤ Fintype.card V) :
    (letI : DecidableRel K.Adj := Classical.decRel K.Adj;
      K.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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