Edge count of an inclusion-maximal drawable graph
ProvedWilliamsonShmoys.maximal_planar_edge_count__p2m_2f30e2d9_4845_4c04_a5c2_ea4608bfb1b9_a1approximation-algorithmsplanar-graphstreewidth
An inclusion-maximal drawable finite simple graph with at least three vertices has exactly 3|V|−6 edges.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn set_option autoImplicit false open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.maximal_planar_edge_count__p2m_2f30e2d9_4845_4c04_a5c2_ea4608bfb1b9_a1
{V : Type} [Fintype V] [LinearOrder V]
(M : SimpleGraph V)
(hV : 3 ≤ Fintype.card V)
(hM : WilliamsonShmoys.HasPlanarDrawingOn M)
(hmax : ∀ N : SimpleGraph V,
Set.Subset M.edgeSet N.edgeSet →
WilliamsonShmoys.HasPlanarDrawingOn N →
N = M) :
M.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).