Connected planar graph edge bounds

Proved

WilliamsonShmoys.connected_planar_edge_bounds__p2m_13cac36b_d337_49ef_981f_d43d139f211c_a2

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

approximation-algorithmsplanar-graphstreewidth

For a finite connected simple graph with at least three vertices and a drawing satisfying HasPlanarDrawingOn, the vertex count is at most one more than the edge count, and the edge count plus six is at most three times the vertex count. The connected bound follows from a spanning tree. The planar bound follows from Euler’s formula and counting incidences along facial boundary walks.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.connected_planar_edge_bounds__p2m_13cac36b_d337_49ef_981f_d43d139f211c_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) [DecidableRel K.Adj]
    (hV : 3 ≤ Fintype.card V)
    (hconn : K.Connected)
    (hK : WilliamsonShmoys.HasPlanarDrawingOn K) :
    Fintype.card V ≤ K.edgeFinset.card + 1 ∧
      K.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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