Planar drawing restricts to subgraphs

Proved

WilliamsonShmoys.planarDrawingOn_subgraph__p2m_3bb112e2_2df9_4e3a_a157_84b2375941f4_a2

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

approximation-algorithmsplanar-graphstreewidth

A subgraph of a graph with a planar drawing on the same ordered vertex type has a planar drawing, using the same positions and edge arcs.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false
open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planarDrawingOn_subgraph__p2m_3bb112e2_2df9_4e3a_a157_84b2375941f4_a2
    {V : Type} [Fintype V] [LinearOrder V]
    {G H : SimpleGraph V} (hHG : H ≤ G)
    (hG : HasPlanarDrawingOn G) : 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).

View graph