A planar supergraph with maximum directed adjacency count

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edge_count_maximal_planar_supergraph__p2m_b64ec28e_2554_405c_95cb_8078e84c56ed_a5

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

approximation-algorithmsplanar-graphstreewidth

Among planar supergraphs of a finite graph, there is one with maximum number of directed adjacent vertex pairs; any planar extension with the same count equals it.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

open WilliamsonShmoys Classical

set_option autoImplicit false

Formal statement

theorem edge_count_maximal_planar_supergraph__p2m_b64ec28e_2554_405c_95cb_8078e84c56ed_a5
    {V : Type} [Fintype V] [LinearOrder V]
    (G : SimpleGraph V)
    (hG : HasPlanarDrawingOn G) :
    ∃ H : SimpleGraph V, G ≤ H ∧ HasPlanarDrawingOn H ∧
      (∀ K : SimpleGraph V, H ≤ K → HasPlanarDrawingOn K →
        Fintype.card {p : V × V // K.Adj p.1 p.2} ≤
          Fintype.card {p : V × V // H.Adj p.1 p.2}) ∧
      (∀ K : SimpleGraph V, H ≤ K → HasPlanarDrawingOn K →
        Fintype.card {p : V × V // H.Adj p.1 p.2} =
          Fintype.card {p : V × V // K.Adj p.1 p.2} → K = H) := 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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