Edge count of a maximal drawable graph
ProvedWilliamsonShmoys.maximal_planar_graph_exact_edge_count__p2m_05cc95b8_6993_44ac_b777_7a2cce137e12_a1approximation-algorithmsplanar-graphstreewidth
An inclusion-maximal drawable simple graph on at least three vertices has exactly 3|V|−6 edges. Prove this by deriving a plane embedding from the supplied crossing-free drawing, showing maximality forces connectedness and triangular faces, and applying Euler’s formula. The hypothesis |V| ≥ 3 is essential to this exact formula and includes the n=3 boundary; it is preserved explicitly. This child contains the remaining topological core obligation.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn set_option autoImplicit false open scoped BigOperators
Formal statement
theorem WilliamsonShmoys.maximal_planar_graph_exact_edge_count__p2m_05cc95b8_6993_44ac_b777_7a2cce137e12_a1
{V : Type} [Fintype V] [LinearOrder V]
(M : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
(hM : WilliamsonShmoys.HasPlanarDrawingOn M)
(hmax : ∀ N : SimpleGraph V, M ≤ N →
WilliamsonShmoys.HasPlanarDrawingOn N → N ≤ M) :
M.edgeSet.ncard = 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).