Maximal planar graph edge lower bound
ProvedWilliamsonShmoys.maximal_planar_edge_lower_bound__p2m_08c24848_031b_43ea_aa26_092fb49494c8_a2approximation-algorithmsplanar-graphstreewidth
A graph with at least three vertices that has a HasPlanarDrawingOn drawing and is maximal among drawable supergraphs on the same vertex type has at least 3|V| - 6 edges. The proof route is to derive connectedness and triangular faces from maximality, then combine Euler’s formula with edge-face incidence counting, including the three-vertex case.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.maximal_planar_edge_lower_bound__p2m_08c24848_031b_43ea_aa26_092fb49494c8_a2
{V : Type} [Fintype V] [LinearOrder V]
(M : SimpleGraph V)
(hcard : 3 <= Fintype.card V)
(hM : HasPlanarDrawingOn M)
(hmax : ∀ N : SimpleGraph V, M <= N -> HasPlanarDrawingOn N -> N = M) :
(letI : (G : SimpleGraph V) -> DecidableRel G.Adj :=
fun G => Classical.decRel G.Adj
3 * Fintype.card V - 6 <= M.edgeFinset.card) := 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).