Edge count of an edge-maximal drawable graph
OpenWilliamsonShmoys.maximal_planar_edge_count__p2m_07b4009a_04ca_4342_a9d5_b274d08c0821_a1approximation-algorithmsplanar-graphstreewidth
If a drawable simple graph on at least three vertices has no drawable strict edge supergraph on those vertices, then it has exactly 3n−6 edges. A proof route is to show maximality forces a connected triangulation, then apply Euler’s formula and count each face boundary edge twice. The n≥3 condition covers the triangle boundary case.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.maximal_planar_edge_count__p2m_07b4009a_04ca_4342_a9d5_b274d08c0821_a1
{V : Type} [Fintype V] [LinearOrder V]
(H : SimpleGraph V) (hH : HasPlanarDrawingOn H)
(hcard : 3 ≤ Fintype.card V) :
letI : ∀ K : SimpleGraph V, DecidableRel K.Adj :=
fun K => Classical.decRel K.Adj
(∀ K : SimpleGraph V,
H.edgeFinset ⊆ K.edgeFinset →
HasPlanarDrawingOn K →
K.edgeFinset ⊆ H.edgeFinset) →
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).