Edge count of an edge-maximal planar graph
OpenWilliamsonShmoys.maximal_planar_graph_edge_count__p2m_16d3ee06_3460_49bf_bee5_c29e2e73cd73_a2approximation-algorithmsplanar-graphstreewidth
An edge-maximal simple graph admitting the specified planar drawing, with at least three vertices, has exactly 3n - 6 edges. The proof route is to establish connectedness and triangular faces, then apply Euler’s formula and face-edge incidence counting.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.maximal_planar_graph_edge_count__p2m_16d3ee06_3460_49bf_bee5_c29e2e73cd73_a2
{V : Type} [Fintype V] [LinearOrder V]
(H : SimpleGraph V) (hH : HasPlanarDrawingOn H)
(hmax : ∀ J : SimpleGraph V, HasPlanarDrawingOn J → H ≤ J → J = H)
(hcard : 3 ≤ Fintype.card V) :
letI : ∀ K : SimpleGraph V, DecidableRel K.Adj :=
fun K => Classical.decRel K.Adj
H.edgeFinset.card = 3 * Fintype.card V - 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).