Edge count of an inclusion-maximal planar graph
OpenWilliamsonShmoys.inclusion_maximal_planar_edge_count__p2m_724aac7b_31c9_4291_9b6d_4edefcb29b4d_a1approximation-algorithmsplanar-graphstreewidth
An inclusion-maximal planar graph on at least three vertices has exactly 3|V| − 6 edges. The n = 3 case is included; maximality forces the graph to be K₃.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.inclusion_maximal_planar_edge_count__p2m_724aac7b_31c9_4291_9b6d_4edefcb29b4d_a1
{V : Type} [Fintype V] [LinearOrder V]
(H : SimpleGraph V) [DecidableRel H.Adj]
(hH : HasPlanarDrawingOn H)
(hcard : 3 ≤ Fintype.card V)
(hmax : ∀ (K : SimpleGraph V) [DecidableRel K.Adj],
HasPlanarDrawingOn K → H ≤ K → K ≤ H) :
H.edgeFinset.card + 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).