Connected planar supergraph extension for nonempty vertex types

Proved

WilliamsonShmoys.planar_drawing_connected_extension__p2m_52ae77ae_f85b_40f5_bd03_34373df951b6_a3

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

approximation-algorithmsplanar-graphstreewidth

For a finite drawn graph whose vertex type has at least three elements, there is a connected simple supergraph on the same vertices that still has a planar drawing. This cardinality condition rules out the empty-type counterexample to the earlier extension statement.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_drawing_connected_extension__p2m_52ae77ae_f85b_40f5_bd03_34373df951b6_a3
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) [DecidableRel K.Adj]
    (hK : HasPlanarDrawingOn K)
    (hcard : 3 <= Fintype.card V) :
    ∃ H : SimpleGraph V, K <= H ∧ H.Connected ∧ 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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