Planar drawings pass to subgraphs

Proved

WilliamsonShmoys.planarDrawingOn_subgraph__p2m_f28d325a_badb_43f7_b5e1_caa5a9a8c825_a2

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

approximation-algorithmsplanar-graphstreewidth

A graph with a planar drawing on the given vertex type passes that drawing to every subgraph on the same vertices.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planarDrawingOn_subgraph__p2m_f28d325a_badb_43f7_b5e1_caa5a9a8c825_a2
    {V : Type} [Fintype V] [LinearOrder V]
    {A B : SimpleGraph V} (hAB : A ≤ B)
    (hB : WilliamsonShmoys.HasPlanarDrawingOn B) :
    WilliamsonShmoys.HasPlanarDrawingOn A := 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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