Edge lower bound for maximal drawable graphs
OpenWilliamsonShmoys.maximal_planar_drawing_edge_lower_bound__p2m_ed0fa66f_dbcb_4794_9398_c71658f4fffe_a1approximation-algorithmsplanar-graphstreewidth
A drawable simple graph on at least three vertices that is inclusion-maximal among drawable graphs 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_drawing_edge_lower_bound__p2m_ed0fa66f_dbcb_4794_9398_c71658f4fffe_a1
{V : Type} [Fintype V] [LinearOrder V]
(G : SimpleGraph V) (hG : HasPlanarDrawingOn G)
(hcard : 3 <= Fintype.card V)
(hmax : forall M : SimpleGraph V, G <= M -> HasPlanarDrawingOn M -> M <= G) :
(letI : DecidableRel G.Adj := Classical.decRel G.Adj;
3 * Fintype.card V <= G.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).