Connected planar supergraph augmentation

Proved

WilliamsonShmoys.planar_drawing_connected_supergraph__p2m_e8818a47_c6bb_451e_9488_1f3faaa47e8f_a3

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

approximation-algorithmsplanar-graphstreewidth

A finite graph with the given crossing-free drawing and at least three vertices has a connected graph on the same vertices that preserves the drawing property and contains at least as many edges. Add edges between components through the complementary regions of the drawing; the proof must formalize this construction for the explicit arc predicate.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_drawing_connected_supergraph__p2m_e8818a47_c6bb_451e_9488_1f3faaa47e8f_a3
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V)
    (hK : HasPlanarDrawingOn K)
    (hcard : 3 ≤ Fintype.card V) :
    (letI : DecidableRel K.Adj := Classical.decRel K.Adj;
      ∃ H : SimpleGraph V,
        HasPlanarDrawingOn H ∧ H.Connected ∧
          K.edgeFinset.card ≤
            (letI : DecidableRel H.Adj := Classical.decRel H.Adj;
              H.edgeFinset.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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