Planar edge augmentation below the triangulation bound
ProvedWilliamsonShmoys.planar_edge_augmentation_below_bound__p2m_8532617a_0690_495a_a2cc_fdaf4080cf05_a1approximation-algorithmsplanar-graphstreewidth
If a finite planar graph on at least three vertices has fewer than 3|V|−6 edges, it has a planar supergraph on the same vertices with at least one added edge. A proof route is to connect components through faces, then add a diagonal in any face whose boundary is not triangular; if no addition is possible, the resulting triangulation and Euler's formula force the edge count to be 3|V|−6. This removes the maximality-to-lower-bound step from the parent proof.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
Formal statement
theorem WilliamsonShmoys.planar_edge_augmentation_below_bound__p2m_8532617a_0690_495a_a2cc_fdaf4080cf05_a1
{V : Type} [Fintype V] [LinearOrder V]
(T : SimpleGraph V)
(hT : WilliamsonShmoys.HasPlanarDrawingOn T)
(hcard : 3 ≤ Fintype.card V)
(hsmall : T.edgeSet.ncard < 3 * Fintype.card V - 6) :
∃ U : SimpleGraph V,
T.edgeSet ⊆ U.edgeSet ∧
∃ e, e ∈ U.edgeSet ∧ e ∉ T.edgeSet ∧
WilliamsonShmoys.HasPlanarDrawingOn U := 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).