Augmenting a disconnected planar graph

Proved

WilliamsonShmoys.edgeMaximalPlanar_augmentDisconnected__p2m_b38e7e0d_55b6_4800_9dde_424996f32dbe_a2

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

approximation-algorithmsplanar-graphstreewidth

Every disconnected finite ordered graph with at least three vertices and a planar drawing has a strict planar supergraph on the same vertices. This isolates the edge-routing argument needed by the parent; the parent then applies maximality to that supergraph.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.edgeMaximalPlanar_augmentDisconnected__p2m_b38e7e0d_55b6_4800_9dde_424996f32dbe_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V)
    (hK : WilliamsonShmoys.HasPlanarDrawingOn K)
    (hn : 3 ≤ Fintype.card V)
    (hdisc : ¬ K.Connected) :
    ∃ H : SimpleGraph V, K ≤ H ∧ H ≠ K ∧ WilliamsonShmoys.HasPlanarDrawingOn H := 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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