Exact edge count of a maximal drawable planar graph
ProvedWilliamsonShmoys.maximal_planar_graph_edge_count__p2m_b501a266_2ceb_4b60_9d31_7e031b204a4c_a1approximation-algorithmsplanar-graphstreewidth
A drawable simple graph on at least three vertices that is maximal under edge inclusion among drawable graphs on the same vertices has exactly 3|V|−6 edges.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
Formal statement
theorem WilliamsonShmoys.maximal_planar_graph_edge_count__p2m_b501a266_2ceb_4b60_9d31_7e031b204a4c_a1
{V : Type} [Fintype V] [LinearOrder V]
(M : SimpleGraph V)
(hM : WilliamsonShmoys.HasPlanarDrawingOn M)
(hcard : 3 ≤ Fintype.card V)
(hMax : ∀ N : SimpleGraph V,
M.edgeSet ⊆ N.edgeSet →
WilliamsonShmoys.HasPlanarDrawingOn N →
N.edgeSet ⊆ M.edgeSet) :
3 * Fintype.card V = M.edgeSet.ncard + 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).