Planar edge bound on each connected component

Proved

WilliamsonShmoys.planar_component_edge_bound__p2m_6875723b_f815_435e_a204_8de0d4b60ca9_a2

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

approximation-algorithmsplanar-graphstreewidth

Every connected component of a finite simple graph with a crossing-free drawing has at most three times its number of vertices minus three edges. The statement installs the finite instance needed for the component vertex subtype and induced graph. For components with at least three vertices, restrict the drawing and apply the connected planar Euler bound; check sizes one and two directly. This removes the component-level geometric argument from the parent, leaving its formalization as the main open obligation.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_component_edge_bound__p2m_6875723b_f815_435e_a204_8de0d4b60ca9_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V)
    (hK : HasPlanarDrawingOn K)
    (c : K.ConnectedComponent) :
    letI : Fintype {v : V // v ∈ c.supp} := Fintype.ofFinite _
    letI : DecidableRel ((K.induce c.supp).Adj) := Classical.decRel _
    (K.induce c.supp).edgeFinset.card ≤
      3 * Fintype.card {v : V // v ∈ c.supp} - 3 := 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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