Component supports partition the vertices

Proved

WilliamsonShmoys.componentwise_edge_bound_aggregate_component_card_sum__p2m_be3c76fd_0a91_4933_b3e9_0bd34bf9ab05_a1

by Eyal1990 · Nov 1, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

The cardinalities of all connected-component supports sum to the number of vertices, because those supports partition the vertex type.

Preamble

import Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected
import Mathlib.Combinatorics.SimpleGraph.Connectivity.Finite
set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.componentwise_edge_bound_aggregate_component_card_sum__p2m_be3c76fd_0a91_4933_b3e9_0bd34bf9ab05_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) [DecidableRel K.Adj] :
    (∑ c : K.ConnectedComponent, c.supp.ncard) = Fintype.card V := by sorry

Source

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).

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