Edge count of an inclusion-maximal drawable graph
OpenWilliamsonShmoys.maximal_planar_graph_edge_count__p2m_9bfda91a_9bda_408b_a2fd_6ab368fb7e28_a1approximation-algorithmsplanar-graphstreewidth
An inclusion-maximal drawable simple graph on at least three vertices has exactly three times the vertex count minus six edges.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib set_option autoImplicit false open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.maximal_planar_graph_edge_count__p2m_9bfda91a_9bda_408b_a2fd_6ab368fb7e28_a1
{V : Type} [Fintype V] [LinearOrder V]
(K : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
(hplanar : HasPlanarDrawingOn K)
(hmax : ∀ J : SimpleGraph V, K.edgeSet ⊆ J.edgeSet →
HasPlanarDrawingOn J → J.edgeSet ⊆ K.edgeSet) :
K.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).