Existence of an edge-maximal planar supergraph

Proved

WilliamsonShmoys.planar_supergraph_edge_maximal__p2m_03cdb16c_d4b2_4cf3_b4f1_33da613ee6f7_a1

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

approximation-algorithmsplanar-graphstreewidth

Among the planar supergraphs of a finite planar graph on the same vertex type, there is one whose edge count is at least that of every other planar supergraph. The proof uses finiteness of the simple graphs on the vertex type; the original graph supplies a candidate.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false
open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_supergraph_edge_maximal__p2m_03cdb16c_d4b2_4cf3_b4f1_33da613ee6f7_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (M : SimpleGraph V) (hV : 3 <= Fintype.card V)
    (hM : HasPlanarDrawingOn M) :
    ∃ N : SimpleGraph V, M ≤ N ∧ HasPlanarDrawingOn N ∧
      ∀ K : SimpleGraph V, M ≤ K → HasPlanarDrawingOn K →
        (letI : DecidableRel K.Adj := Classical.decRel K.Adj
         letI : DecidableRel N.Adj := Classical.decRel N.Adj
         K.edgeFinset.card ≤ N.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).

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