Euler edge bound for a connected planar supergraph
ProvedWilliamsonShmoys.connected_planar_edge_bound__p2m_b6cbb51d_e0ab_4ed8_a943_b0435eeee10f_a3approximation-algorithmsplanar-graphstreewidth
If a finite simple graph M is a subgraph of a connected graph H with a crossing-free drawing, and there are at least three vertices, then M has at most 3|V|−6 edges. This removes the parent’s edge-counting obligation: prove Euler’s formula for the connected planar drawing, count two edge incidences per face, and use simplicity and at least three vertices to bound face lengths below by three; then use M ≤ H to transfer the edge bound. The size hypothesis is essential to this Euler bound and is preserved explicitly. The parent follows by applying the augmentation child to obtain H and then this bound.
Preamble
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn set_option autoImplicit false open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.connected_planar_edge_bound__p2m_b6cbb51d_e0ab_4ed8_a943_b0435eeee10f_a3
{V : Type} [Fintype V] [LinearOrder V]
(M H : SimpleGraph V)
(hMH : M ≤ H)
(hHconn : ∀ u v, Relation.ReflTransGen H.Adj u v)
(hHdraw : WilliamsonShmoys.HasPlanarDrawingOn H)
(hV : 3 ≤ Fintype.card V) :
M.edgeSet.ncard + 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).