Strict graph extension increases directed arc count

Proved

WilliamsonShmoys.arc_count_strict_of_graph_strict__p2m_c50204fa_335d_4575_bac2_23cf83d1b5e4_a2

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

approximation-algorithmsplanar-graphstreewidth

For a finite vertex type, a proper simple-graph extension strictly increases the number of directed adjacent vertex pairs.

Preamble

import Mathlib
import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.arc_count_strict_of_graph_strict__p2m_c50204fa_335d_4575_bac2_23cf83d1b5e4_a2
    {V : Type} [Fintype V]
    (K L : SimpleGraph V) (hKL : K ≤ L) (hne : K ≠ L) :
    Nat.card {p : V × V // K.Adj p.1 p.2} <
      Nat.card {p : V × V // L.Adj p.1 p.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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