Edge bound for a disconnected finite planar graph

Proved

WilliamsonShmoys.planar_graph_edge_bound_disconnected__p2m_e05e843e_1b16_4381_acec_2ded01ba2eda_a2

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

approximation-algorithmsplanar-graphstreewidth

A disconnected finite simple graph with at least three vertices and a crossing-free drawing satisfies the planar edge bound. Restrict the drawing to each connected component, use the connected estimate on components with at least three vertices, bound the one- and two-vertex components directly, and sum the component edge counts.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_graph_edge_bound_disconnected__p2m_e05e843e_1b16_4381_acec_2ded01ba2eda_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V)
    (hcard : 3 ≤ Fintype.card V)
    (hconn : ¬ H.Connected)
    (hdraw : WilliamsonShmoys.HasPlanarDrawingOn H) :
    (letI : DecidableRel H.Adj := Classical.decRel H.Adj
     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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