Edge lower bound from planar maximality
ProvedWilliamsonShmoys.maximal_planar_edge_count_lower_bound__p2m_77901df4_aad7_410f_861f_76a2bd150be4_a1approximation-algorithmsplanar-graphstreewidth
A drawable finite simple graph on at least three vertices that admits no drawable supergraph on the same vertices has at least three times the number of vertices minus six edges. The intended route uses maximality to force a connected triangulation and then counts face-edge incidences with Euler’s formula.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.maximal_planar_edge_count_lower_bound__p2m_77901df4_aad7_410f_861f_76a2bd150be4_a1
{V : Type} [Fintype V] [LinearOrder V]
(G : SimpleGraph V) (hV : 3 <= Fintype.card V)
(hG : WilliamsonShmoys.HasPlanarDrawingOn G)
(hmax : ∀ L : SimpleGraph V, G <= L → WilliamsonShmoys.HasPlanarDrawingOn L → L <= G) :
letI : DecidableRel G.Adj := Classical.decRel G.Adj
3 * Fintype.card V <= G.edgeFinset.card + 6 := 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).