Derive the simple planar edge bound from face data
ProvedWilliamsonShmoys.planarMapData_edge_bound__p2m_b23772a1_7f55_4be6_b745_ebb5a4e685d3_a4approximation-algorithmsplanar-graphstreewidth
Face data recording Euler’s equation, total boundary incidence twice the edge count, and degree at least three for every face imply the simple planar edge bound.
Preamble
import Definitions.Def_PlanarMapData__p2m_b23772a1_7f55_4be6_b745_ebb5a4e685d3_a3 import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib
Formal statement
theorem WilliamsonShmoys.planarMapData_edge_bound__p2m_b23772a1_7f55_4be6_b745_ebb5a4e685d3_a4
{V : Type} [Fintype V]
(J : SimpleGraph V) [DecidableRel J.Adj]
(D : PlanarMapData__p2m_b23772a1_7f55_4be6_b745_ebb5a4e685d3_a3 J) :
J.edgeFinset.card + 6 <= 3 * Fintype.card 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).