Edge count detects equality under graph inclusion
ProvedWilliamsonShmoys.SimpleGraph.edge_card_mono_and_eq_of_le__p2m_8bdcd24b_0a33_4788_8eda_5b06fcc2996a_a2approximation-algorithmsplanar-graphstreewidth
For finite-vertex simple graphs, inclusion implies edge-count monotonicity, and inclusion together with the reverse edge-count inequality implies graph equality. In the parent, this proves that H has no more edges than the chosen supergraph K, and that any drawable L extending K with no greater edge count must equal K. A proof route uses inclusion of edge finsets, monotonicity of finite-set cardinality, and equality of finite sets with equal cardinality under inclusion.
Preamble
import Mathlib
Formal statement
theorem WilliamsonShmoys.SimpleGraph.edge_card_mono_and_eq_of_le__p2m_8bdcd24b_0a33_4788_8eda_5b06fcc2996a_a2
{V : Type} [Fintype V]
(G H : SimpleGraph V) (hGH : G ≤ H) :
(letI : DecidableRel G.Adj := Classical.decRel G.Adj
letI : DecidableRel H.Adj := Classical.decRel H.Adj
G.edgeFinset.card ≤ H.edgeFinset.card ∧
(H.edgeFinset.card ≤ G.edgeFinset.card → G = H)) := 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).