Euler face count for a connected planar drawing

Proved

WilliamsonShmoys.connected_planar_euler_face_count__p2m_b2f31750_6395_4f01_b4dd_f5215a908d6d_a1

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

approximation-algorithmsplanar-graphstreewidth

For a finite connected graph with the supplied planar drawing, there is a natural face count f satisfying Euler’s formula. The connectedness hypothesis is essential: without it, the face count also depends on the number of connected components. A proof route is to extract a finite plane embedding from the drawing and apply Euler’s formula for connected embeddings.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false

Formal statement

theorem WilliamsonShmoys.connected_planar_euler_face_count__p2m_b2f31750_6395_4f01_b4dd_f5215a908d6d_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (N : SimpleGraph V)
    (hN : WilliamsonShmoys.HasPlanarDrawingOn N)
    (hconn : N.Connected) :
    letI : DecidableRel N.Adj := Classical.decRel N.Adj
    ∃ f : ℕ, Fintype.card V + f = N.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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