Edge lower bound for a maximal finite topological planar drawing
ProvedWilliamsonShmoys.planar_edge_count_lower__p2m_88f6e318_47fb_49e3_83e2_88b806f3a0ca_a1approximation-algorithmsplanar-graphstreewidth
A maximal graph among those satisfying HasPlanarDrawingOn, with at least three vertices, has at least 3|V| − 6 edges. The proof must show that maximality forces a triangulated plane embedding, then apply Euler’s formula and count face-edge incidences.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.planar_edge_count_lower__p2m_88f6e318_47fb_49e3_83e2_88b806f3a0ca_a1
{V : Type} [Fintype V] [LinearOrder V]
(H : SimpleGraph V) (hV : 3 <= Fintype.card V)
(hH : HasPlanarDrawingOn H)
(hmax : ∀ K : SimpleGraph V, H ≤ K → HasPlanarDrawingOn K → K = H) :
(letI : DecidableRel H.Adj := Classical.decRel H.Adj;
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).