Finite maximality yields a connected extension
ProvedWilliamsonShmoys.finite_graph_maximal_extension__p2m_1743449e_70b2_453d_82e4_df1f7d1e75bd_a3approximation-algorithmsplanar-graphstreewidth
On a finite vertex type, if every disconnected graph extending G and satisfying Q has a strictly larger extension still satisfying Q, then some connected extension of G satisfies Q. Prove this by maximizing edge-set cardinality among eligible graphs.
Preamble
import Mathlib set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.finite_graph_maximal_extension__p2m_1743449e_70b2_453d_82e4_df1f7d1e75bd_a3
{V : Type} [Fintype V] [DecidableEq V]
(G : SimpleGraph V) (Q : SimpleGraph V → Prop) (hQ : Q G)
(hstep : ∀ (K : SimpleGraph V),
G.edgeSet ⊆ K.edgeSet → Q K → ¬ K.Connected →
∃ L : SimpleGraph V, K.edgeSet ⊂ L.edgeSet ∧ Q L) :
∃ H : SimpleGraph V,
G.edgeSet ⊆ H.edgeSet ∧ Q H ∧ H.Connected := 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).