Strict simple graph supergraphs add an edge
ProvedWilliamsonShmoys.strict_simplegraph_supergraph_has_new_edge__p2m_b7a1cfc0_b36e_4034_a5fa_2e691bcea0a4_a1approximation-algorithmsplanar-graphstreewidth
If one simple graph includes another but is not equal to it, then it has an edge absent from the smaller graph.
Preamble
import Mathlib
Formal statement
theorem WilliamsonShmoys.strict_simplegraph_supergraph_has_new_edge__p2m_b7a1cfc0_b36e_4034_a5fa_2e691bcea0a4_a1
{V : Type} [Fintype V]
(K L : SimpleGraph V) (hsub : K ≤ L) (hne : L ≠ K) :
∃ u v, L.Adj u v ∧ ¬ K.Adj u 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).