Edge count is monotone under graph inclusion
ProvedWilliamsonShmoys.edgeFinset_card_mono__p2m_df5daf12_daf3_4429_a1d3_fb22e1c60853_a1approximation-algorithmsplanar-graphstreewidth
For finite graphs on the same vertex type, inclusion of adjacency relations implies inclusion of their edge finsets and hence monotonicity of edge count. This isolates the finite-set step used after maximality gives K ≤ H. It can be proved from Mathlib’s edge-finset inclusion theorem and Finset.card_le_card.
Preamble
import Mathlib
Formal statement
theorem WilliamsonShmoys.edgeFinset_card_mono__p2m_df5daf12_daf3_4429_a1d3_fb22e1c60853_a1
{V : Type} [Fintype V]
(G F : SimpleGraph V) [DecidableRel G.Adj] [DecidableRel F.Adj]
(hGF : G ≤ F) :
G.edgeFinset.card ≤ F.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).