Proper simple graph inclusion strictly increases edge count
Provedfinite_graph_property_maximal_supergraph_strict_edge_count__p2m_98fc4204_1185_42f8_9d24_9b96a2776006_a1approximation-algorithmsplanar-graphstreewidth
On a finite vertex type, if one simple graph is edge-included in another and the latter has an edge missing from the former, then the latter has strictly more edges.
Preamble
import Mathlib
Formal statement
theorem finite_graph_property_maximal_supergraph_strict_edge_count__p2m_98fc4204_1185_42f8_9d24_9b96a2776006_a1
{V : Type} [Fintype V]
(H J : SimpleGraph V)
(dH : DecidableRel H.Adj) (dJ : DecidableRel J.Adj)
(hHJ : ∀ u v, H.Adj u v → J.Adj u v)
(hstrict : ∃ u v, J.Adj u v ∧ ¬ H.Adj u v) :
(letI : DecidableRel H.Adj := dH
letI : DecidableRel J.Adj := dJ
H.edgeFinset.card < J.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).