Graph equality from inclusion and adjacent-pair count
ProvedWilliamsonShmoys.simpleGraph_eq_of_le_of_adjPairCard_le__p2m_989b9ffa_ac83_4205_b86c_4def2f85e45f_a3approximation-algorithmsplanar-graphstreewidth
For a finite vertex type, if H is a subgraph of K and K has no more ordered adjacent pairs than H, then the graphs are equal.
Preamble
import Mathlib set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.simpleGraph_eq_of_le_of_adjPairCard_le__p2m_989b9ffa_ac83_4205_b86c_4def2f85e45f_a3
{V : Type} [Fintype V]
(H K : SimpleGraph V)
(hHK : H ≤ K)
(hcard : Nat.card {p : V × V // K.Adj p.1 p.2} ≤
Nat.card {p : V × V // H.Adj p.1 p.2}) :
K = 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).