Monotonicity of edge count under graph inclusion
ProvedWilliamsonShmoys.edge_card_mono__p2m_7421bb86_5406_4559_80b0_0937bd98493b_a2approximation-algorithmsplanar-graphstreewidth
If G is a subgraph of K on the same finite vertex type, its edge count is no larger. The proof route uses inclusion of the corresponding edge finsets and cardinality monotonicity. The statement retains the classical adjacency decision instances required by edgeFinset.
Preamble
import Mathlib set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.edge_card_mono__p2m_7421bb86_5406_4559_80b0_0937bd98493b_a2 {V : Type} [Fintype V] (G K : SimpleGraph V) (hGK : G ≤ K) : (letI : DecidableRel G.Adj := Classical.decRel G.Adj; letI : DecidableRel K.Adj := Classical.decRel K.Adj; G.edgeFinset.card ≤ K.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).