Extract vertex and edge statistics of planar connected components

Proved

WilliamsonShmoys.component_statistics__p2m_24514a04_46c3_4995_8d84_ff88ef385ed2_a2

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

approximation-algorithmsplanar-graphstreewidth

For a finite drawn graph, produce a list of connected components’ vertex and edge counts that sums to the graph totals. Components of size at least three satisfy the assumed connected estimate; smaller components have at most as many edges as vertices.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.component_statistics__p2m_24514a04_46c3_4995_8d84_ff88ef385ed2_a2
    {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) :
    ∃ parts : List (Nat × Nat),
      (parts.map Prod.fst).sum = Fintype.card V ∧
      (parts.map Prod.snd).sum = M.edgeFinset.card ∧
      ∀ p ∈ parts,
        (3 ≤ p.1 → p.2 ≤ 3 * p.1 - 6) ∧
        (p.1 < 3 → p.2 ≤ p.1) := 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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