Edge lower bound for an edge-maximal planar graph
ProvedWilliamsonShmoys.maximal_planar_edge_lower_bound__p2m_2629fd8d_6fc9_4c07_8b09_9c33cf4a1063_a2approximation-algorithmsplanar-graphstreewidth
Every edge-maximal simple planar graph on at least three vertices has at least 3|V| − 6 edges.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.maximal_planar_edge_lower_bound__p2m_2629fd8d_6fc9_4c07_8b09_9c33cf4a1063_a2 {V : Type} [Fintype V] [LinearOrder V] (H : SimpleGraph V) [DecidableRel H.Adj] (hH : HasPlanarDrawingOn H) (hcard : 3 ≤ Fintype.card V) (hmax : ∀ K : SimpleGraph V, (∀ u v, H.Adj u v → K.Adj u v) → HasPlanarDrawingOn K → ∀ u v, K.Adj u v → H.Adj u v) : 3 * Fintype.card V ≤ H.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).