Restrict a planar drawing to a subgraph

Proved

WilliamsonShmoys.planar_drawing_restrict_subgraph__p2m_6d91b10d_a30f_4f1a_8acb_96a7ed12e6b9_a1

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

approximation-algorithmsplanar-graphstreewidth

If M is a subgraph of N and N has the specified crossing-free drawing, the same vertex placement and edge arcs give a drawing of M.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn

open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_drawing_restrict_subgraph__p2m_6d91b10d_a30f_4f1a_8acb_96a7ed12e6b9_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (M N : SimpleGraph V) (hMN : M ≤ N)
    (hNdraw : WilliamsonShmoys.HasPlanarDrawingOn N) :
    WilliamsonShmoys.HasPlanarDrawingOn M := 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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