Simple planar graph edge bound
ProvedWilliamsonShmoys.simple_planar_edge_bound__p2m_09878566_3989_4e48_bc10_98e236546b3d_a1__p2m_09878566_3989_4e48_bc10_98e236546b3d_a2approximation-algorithmsplanar-graphstreewidth
A simple connected graph with the specified planar drawing and at least three vertices has at most three times its vertex count minus six edges, expressed without natural-number subtraction. The size hypothesis retains the boundary condition needed for the planar edge bound.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected set_option autoImplicit false open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.simple_planar_edge_bound__p2m_09878566_3989_4e48_bc10_98e236546b3d_a1__p2m_09878566_3989_4e48_bc10_98e236546b3d_a2
{V : Type} [Fintype V] [LinearOrder V]
(G : SimpleGraph V) (hcard : 3 <= Fintype.card V)
(hconn : G.Connected)
(hG : WilliamsonShmoys.HasPlanarDrawingOn G) :
G.edgeSet.ncard + 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).