Component vertex and edge accounting
ProvedWilliamsonShmoys.component_vertex_accounting__p2m_ddc73965_e52c_4f89_a584_0b5c76884424_a1__p2m_ddc73965_e52c_4f89_a584_0b5c76884424_a3approximation-algorithmsplanar-graphstreewidth
The supports of connected components partition the vertices, and the induced edge sets of those supports partition the graph’s edges. Therefore their vertex and edge cardinalities sum to the corresponding graph counts, and each support is nonempty.
Preamble
import Mathlib set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.component_vertex_accounting__p2m_ddc73965_e52c_4f89_a584_0b5c76884424_a1__p2m_ddc73965_e52c_4f89_a584_0b5c76884424_a3 {V : Type} [Fintype V] [LinearOrder V] (L : SimpleGraph V) [Fintype L.ConnectedComponent] : (Finset.univ.sum fun c : L.ConnectedComponent => (SimpleGraph.ConnectedComponent.supp c).ncard) = Fintype.card V ∧ (Finset.univ.sum fun c : L.ConnectedComponent => (L.induce (SimpleGraph.ConnectedComponent.supp c)).edgeSet.ncard) = L.edgeSet.ncard ∧ (∀ c : L.ConnectedComponent, 1 ≤ (SimpleGraph.ConnectedComponent.supp c).ncard) := 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).