Edge bound for connected drawn planar graphs

Proved

WilliamsonShmoys.connected_planar_drawing_edge_bound__p2m_52ae77ae_f85b_40f5_bd03_34373df951b6_a2

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

approximation-algorithmsplanar-graphstreewidth

This new lemma isolates the main topological work: the connected case of the target. Convert the supplied drawing into its plane embedding, use Euler’s formula V − E + F = 2, and count face-edge incidences. Each edge has two sides, and for a connected simple graph with at least three vertices each facial boundary walk has length at least three; thus 3F ≤ 2E, yielding E + 6 ≤ 3V. The connectedness, simplicity, and vertex-count hypotheses are retained to exclude the exceptional small facial walks.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.connected_planar_drawing_edge_bound__p2m_52ae77ae_f85b_40f5_bd03_34373df951b6_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V) [DecidableRel H.Adj]
    (hH : HasPlanarDrawingOn H)
    (hconn : H.Connected)
    (hcard : 3 ≤ Fintype.card V) :
    H.edgeFinset.card + 6 ≤ 3 * Fintype.card V := 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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