Planar drawings restrict to subgraphs

Proved

WilliamsonShmoys.planarDrawingOn_subgraph_closed__p2m_813de8ea_e58e_48bb_807f_43145eda7132_a1

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

approximation-algorithmsplanar-graphstreewidth

Any subgraph of a graph with a crossing-free drawing on the same vertex set also has such a drawing.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib
open WilliamsonShmoys
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planarDrawingOn_subgraph_closed__p2m_813de8ea_e58e_48bb_807f_43145eda7132_a1 {V : Type} [Fintype V] [LinearOrder V] {G H : SimpleGraph V} (hGH : G ≤ H) (hH : HasPlanarDrawingOn H) : HasPlanarDrawingOn G := 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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