Connected planar supergraph with at least three vertices

Proved

WilliamsonShmoys.planar_drawing_connected_supergraph_of_three_vertices__p2m_4d9d61b3_a3d4_4adb_a77f_fb6a2cb8e8dc_a2

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

approximation-algorithmsplanar-graphstreewidth

For a finite ordered vertex type with at least three vertices, every graph with a crossing-free drawing has a connected drawable supergraph on the same vertices. Existing edges and vertices are preserved; the size hypothesis excludes the empty-vertex counterexample to the earlier unrestricted statement.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_drawing_connected_supergraph_of_three_vertices__p2m_4d9d61b3_a3d4_4adb_a77f_fb6a2cb8e8dc_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V)
    (hK : WilliamsonShmoys.HasPlanarDrawingOn K)
    (hn : 3 ≤ Fintype.card V) :
    ∃ H : SimpleGraph V,
      K ≤ H ∧ H.Connected ∧ 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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