Extract finite face data from a planar drawing
ProvedWilliamsonShmoys.planarDrawingOn_exists_mapData__p2m_b23772a1_7f55_4be6_b745_ebb5a4e685d3_a3approximation-algorithmsplanar-graphstreewidth
A connected simple graph with at least three vertices and the supplied crossing-free drawing admits finite face data recording Euler's equation, twice-per-edge boundary incidence, and degree at least three for every face. This child carries the geometric construction that is not represented by the supplied drawing definition.
Preamble
import Definitions.Def_PlanarMapData__p2m_b23772a1_7f55_4be6_b745_ebb5a4e685d3_a3 import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.planarDrawingOn_exists_mapData__p2m_b23772a1_7f55_4be6_b745_ebb5a4e685d3_a3
{V : Type} [Fintype V] [LinearOrder V]
(J : SimpleGraph V) [DecidableRel J.Adj]
(hJ : HasPlanarDrawingOn J)
(hconn : forall u v : V, Nonempty (J.Walk u v))
(hcard : 3 <= Fintype.card V) :
Nonempty (PlanarMapData__p2m_b23772a1_7f55_4be6_b745_ebb5a4e685d3_a3 J) := 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).