Euler and face-incidence counts for maximal planar drawings

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WilliamsonShmoys.maximal_planar_euler_face_count__p2m_1927a2f2_efe0_4e83_ba03_5beb334dbd6d_a1

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

approximation-algorithmsplanar-graphstreewidth

For a finite, linearly ordered vertex type with at least three vertices, a graph with the stated drawing that is maximal among drawable supergraphs has a face count satisfying Euler’s formula and the triangular-face incidence count.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.maximal_planar_euler_face_count__p2m_1927a2f2_efe0_4e83_ba03_5beb334dbd6d_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (K : SimpleGraph V) (hcard : 3 ≤ Fintype.card V)
    (hplanar : HasPlanarDrawingOn K)
    (hmax : ∀ L : SimpleGraph V,
      K.edgeSet ⊆ L.edgeSet →
      HasPlanarDrawingOn L →
      L.edgeSet = K.edgeSet) :
    ∃ f : ℕ,
      f + Fintype.card V = K.edgeSet.ncard + 2 ∧
      3 * f = 2 * K.edgeSet.ncard := 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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