Edge bound for a connected planar induced subgraph

Proved

WilliamsonShmoys.connectedPlanarInducedEdgeBound__p2m_d29cc926_0e2e_4704_bc36_256caa004206_a1

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

approximation-algorithmsplanar-graphstreewidth

For any subset of a finite drawn planar graph whose induced subgraph is connected, the number of induced edges plus three is at most three times the subset size, including the one- and two-vertex cases.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.connectedPlanarInducedEdgeBound__p2m_d29cc926_0e2e_4704_bc36_256caa004206_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (L : SimpleGraph V) (dL : DecidableRel L.Adj)
    (hL : HasPlanarDrawingOn L)
    (s : Finset V) (hconn : (L.induce (↑s : Set V)).Connected) :
    (letI : DecidableRel L.Adj := dL
     (L.induce (↑s : Set V)).edgeFinset.card) + 3 ≤ 3 * s.card := 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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