A strict planar edge-count gap forces a strict extension

Proved

strict_supergraph_of_planar_edge_count_gap__p2m_d8aa4474_1bb0_4a8a_92a2_795d49f289c9_a1

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

approximation-algorithmsplanar-graphstreewidth

If $K$ is a subgraph of $L$ , the edge count of $K$ is strictly below the triangulation count, and $L$ has that count, then $L$ is not a subgraph of $K$ .

Preamble

import Mathlib

Formal statement

theorem strict_supergraph_of_planar_edge_count_gap__p2m_d8aa4474_1bb0_4a8a_92a2_795d49f289c9_a1
    {V : Type} [Fintype V] [LinearOrder V]
    (K L : SimpleGraph V)
    (dK : DecidableRel K.Adj)
    (hKL : K ≤ L)
    (hsmall : (letI : DecidableRel K.Adj := dK
      K.edgeFinset.card < 3 * Fintype.card V - 6))
    (hcount : (letI : DecidableRel L.Adj := Classical.decRel L.Adj
      L.edgeFinset.card = 3 * Fintype.card V - 6)) :
    ¬ L ≤ K := 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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