Lower bound on total face-boundary length

Proved

WilliamsonShmoys.planar_face_count_boundary_lower_bound__p2m_ce2523b8_2817_4847_92d2_52162bbbf3f4_a1

by Eyal1990 · Oct 31, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

For the face count of a connected simple graph with at least three vertices and the supplied crossing-free drawing, the sum of face-boundary lengths is at least three per face and equals twice the number of edges. Thus three times the face count is at most twice the edge count.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.planar_face_count_boundary_lower_bound__p2m_ce2523b8_2817_4847_92d2_52162bbbf3f4_a1 {V : Type} [Fintype V] [LinearOrder V] (M : SimpleGraph V) [DecidableRel M.Adj] (hM : WilliamsonShmoys.HasPlanarDrawingOn M) (hconn : M.Connected) (hcard : 3 ≤ Fintype.card V) (f : ℕ) (hEuler : Fintype.card V + f = M.edgeFinset.card + 2) : 3 * f ≤ 2 * M.edgeFinset.card := 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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