Connected planar drawing yields facial boundary data and Euler’s equation

Proved

WilliamsonShmoys.connected_planar_face_certificate__p2m_42c50023_969e_4d67_9641_cde1dc4669ab_a2

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

approximation-algorithmsplanar-graphstreewidth

Construct a finite face count and boundary lengths from the explicit drawing. Their sum counts each edge twice, each face boundary has length at least three, and the vertex, face, and edge counts satisfy Euler’s equation. Connectedness is needed for this Euler relation; simplicity and the lower bound on vertices support the boundary-length condition, including in the presence of bridges.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.connected_planar_face_certificate__p2m_42c50023_969e_4d67_9641_cde1dc4669ab_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) [DecidableRel K.Adj]
    (hK : HasPlanarDrawingOn K)
    (hconn : K.Connected)
    (hcard : 3 ≤ Fintype.card V) :
    ∃ faceCount : Nat,
      (∃ boundaryLength : Fin faceCount → Nat,
        (∑ f : Fin faceCount, boundaryLength f) = 2 * K.edgeFinset.card ∧
        (∀ f : Fin faceCount, 3 ≤ boundaryLength f)) ∧
      Fintype.card V + faceCount = K.edgeFinset.card + 2 := 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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