Extend a connected planar drawing to a maximal planar graph
ProvedWilliamsonShmoys.planar_supergraph_maximal__p2m_3c9d29e6_223d_45fd_8079_57d4a2d204ad_a1approximation-algorithmsplanar-graphstreewidth
A connected graph with at least three vertices and a drawing satisfying HasPlanarDrawingOn has a maximal planar supergraph on the same vertex set. Maximality means every planar supergraph on that vertex set is equal to it.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.planar_supergraph_maximal__p2m_3c9d29e6_223d_45fd_8079_57d4a2d204ad_a1
{V : Type} [Fintype V] [LinearOrder V]
(L : SimpleGraph V)
(hcard : 3 <= Fintype.card V)
(hconn : L.Connected)
(hL : HasPlanarDrawingOn L) :
∃ M : SimpleGraph V, L <= M ∧ HasPlanarDrawingOn M ∧
(∀ N : SimpleGraph V, M <= N → HasPlanarDrawingOn N → N = M) := 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).