Restrict a planar drawing to an induced subgraph

Proved

WilliamsonShmoys.planar_drawingOn_induce__p2m_ee19bce3_aed2_4043_9cb4_9bf993438165_a1

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

approximation-algorithmsplanar-graphstreewidth

Restricting the vertex placement and edge arcs of a drawing to the vertices in a finite subset gives a drawing, in the same sense, of the induced subgraph. This supplies the drawing hypothesis needed when applying the connected bound to each connected induced component.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_drawingOn_induce__p2m_ee19bce3_aed2_4043_9cb4_9bf993438165_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V) (hG : WilliamsonShmoys.HasPlanarDrawingOn G)
    (s : Finset V) :
    WilliamsonShmoys.HasPlanarDrawingOn (G.induce (↑s : Set V)) := 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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