Planar graphs admit a triangulated planar supergraph
ProvedWilliamsonShmoys.planar_triangulation_completion__p2m_34ce151f_05d3_497c_a88f_b630e2f0d090_a1__p2m_34ce151f_05d3_497c_a88f_b630e2f0d090_a2approximation-algorithmsplanar-graphstreewidth
Every finite simple graph with at least three vertices that has a crossing-free drawing on the given vertex type is contained in a planar supergraph on that same vertex type with exactly 3|V|−6 edges. This includes disconnected graphs.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.planar_triangulation_completion__p2m_34ce151f_05d3_497c_a88f_b630e2f0d090_a1__p2m_34ce151f_05d3_497c_a88f_b630e2f0d090_a2
{V : Type} [Fintype V] [LinearOrder V]
(K : SimpleGraph V) (hK : HasPlanarDrawingOn K)
(hn : 3 <= Fintype.card V) :
Exists fun L : SimpleGraph V => K <= L /\ HasPlanarDrawingOn L /\
(letI : DecidableRel L.Adj := Classical.decRel _;
L.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).