Drawable graphs admit an Euler-count extension
ProvedWilliamsonShmoys.drawable_graph_has_euler_extension__p2m_8d2492c2_a620_4e30_8a73_7621e3b1e96d_a2approximation-algorithmsplanar-graphstreewidth
Every drawable simple graph on a finite linearly ordered vertex type with at least three vertices is a subgraph of a drawable graph on the same vertices whose edge count is exactly three times the number of vertices minus six.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn import Mathlib open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.drawable_graph_has_euler_extension__p2m_8d2492c2_a620_4e30_8a73_7621e3b1e96d_a2
{V : Type} [Fintype V] [LinearOrder V]
(K : SimpleGraph V) (hV : 3 ≤ Fintype.card V)
(hK : HasPlanarDrawingOn K) :
∃ L : SimpleGraph V, K ≤ L ∧ HasPlanarDrawingOn L ∧
(letI : DecidableRel L.Adj := Classical.decRel L.Adj
L.edgeFinset.card + 6 = 3 * Fintype.card V) := 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).