Euler and triangular-face counts for maximal planar drawings

Open

WilliamsonShmoys.maximal_planar_face_count__p2m_6bb0623d_4ea1_4aa8_a0da_a55490a47412_a2

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

approximation-algorithmsplanar-graphstreewidth

If H has at least three vertices, is connected, has the exact crossing-free arc drawing specified by HasPlanarDrawingOn, and is inclusion-maximal among drawable supergraphs on the same vertices, then there is a natural number f satisfying Euler’s equation n + f = m + 2 and the triangular face-incidence equation 3f = 2m, where m is the number of edges.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.maximal_planar_face_count__p2m_6bb0623d_4ea1_4aa8_a0da_a55490a47412_a2
    {V : Type} [Fintype V] [LinearOrder V]
    (H : SimpleGraph V)
    (hcard : 3 ≤ Fintype.card V)
    (hconn : H.Connected)
    (hdraw : WilliamsonShmoys.HasPlanarDrawingOn H)
    (hmax : ∀ K : SimpleGraph V,
      H ≤ K → WilliamsonShmoys.HasPlanarDrawingOn K → K ≤ H) :
    (letI : DecidableRel H.Adj := Classical.decRel H.Adj
     ∃ f : ℕ,
       Fintype.card V + f = H.edgeFinset.card + 2 ∧
       3 * f = 2 * H.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).

View graph