Combine connected-component edge bounds

Proved

WilliamsonShmoys.planar_graph_edge_upper_bound_of_components__p2m_d428652f_4806_46f7_95cc_5b75c77bec4b_a1

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

approximation-algorithmsplanar-graphstreewidth

If the edge bound holds for every connected finite simple graph with a drawing, then it holds for any finite simple graph with a drawing and at least three vertices. Restrict the drawing to connected components, use the connected bound on components of size at least three, handle components of size one or two directly, and sum their edge counts and vertex counts.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_graph_edge_upper_bound_of_components__p2m_d428652f_4806_46f7_95cc_5b75c77bec4b_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (hconnected : ∀ {U : Type} [Fintype U] [LinearOrder U]
      (N : SimpleGraph U) (dN : DecidableRel N.Adj),
      HasPlanarDrawingOn N → N.Connected → 3 ≤ Fintype.card U →
      N.edgeFinset.card ≤ 3 * Fintype.card U - 6)
    (M : SimpleGraph V) (dM : DecidableRel M.Adj)
    (hM : HasPlanarDrawingOn M)
    (hcard : 3 ≤ Fintype.card V) :
    M.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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