Planar drawing is preserved by taking subgraphs

Proved

hasPlanarDrawingOn_mono__p2m_03bfe837_ddc7_49ff_ba57_41dcefd0c29f_a1

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

approximation-algorithmsplanar-graphstreewidth

If a graph on a fixed finite ordered vertex type has a crossing-free drawing, every subgraph on the same vertex type also has a crossing-free drawing.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

Formal statement

theorem hasPlanarDrawingOn_mono__p2m_03bfe837_ddc7_49ff_ba57_41dcefd0c29f_a1
    {V : Type} [Fintype V] [LinearOrder V]
    {G H : SimpleGraph V}
    (hHG : H ≤ G)
    (hG : WilliamsonShmoys.HasPlanarDrawingOn G) :
    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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