Add an edge between disconnected components of a planar drawing
OpenWilliamsonShmoys.planar_drawing_add_edge_of_unreachable__p2m_f0a53f14_3d07_482d_8483_9c70fdc17ecd_a1approximation-algorithmsplanar-graphstreewidth
Given a finite ordered planar graph and unreachable vertices, the graph obtained by adding the edge between those vertices has a crossing-free drawing of the specified kind. The unreachable hypothesis ensures the added edge joins distinct components and is not already present; the proof must preserve all original drawing conditions.
Preamble
import Mathlib import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.planar_drawing_add_edge_of_unreachable__p2m_f0a53f14_3d07_482d_8483_9c70fdc17ecd_a1
{V : Type} [Fintype V] [LinearOrder V]
(H : SimpleGraph V) (hH : HasPlanarDrawingOn H)
(u v : V) (hnot : ¬ H.Reachable u v) :
HasPlanarDrawingOn (H ⊔ SimpleGraph.fromEdgeSet {s(u, v)}) := by sorrySource
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).