Subgraphs have no more edges
ProvedWilliamsonShmoys.subgraph_edgeFinset_card_mono__p2m_0c1d2833_de32_49c5_aebf_88ae0a9260a2_a1approximation-algorithmsplanar-graphstreewidth
If G is a subgraph of H on a finite vertex type, its edge finset has cardinality at most that of H, without a vertex-count restriction.
Preamble
import Mathlib set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.subgraph_edgeFinset_card_mono__p2m_0c1d2833_de32_49c5_aebf_88ae0a9260a2_a1
{V : Type} [Fintype V]
(G H : SimpleGraph V) [DecidableRel G.Adj] [DecidableRel H.Adj]
(hsub : G ≤ H) :
G.edgeFinset.card ≤ H.edgeFinset.card := 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).