Finite face data for a plane graph
DefinitionPlanarMapData__p2m_b23772a1_7f55_4be6_b745_ebb5a4e685d3_a3approximation-algorithmsplanar-graphstreewidth
A finite record of face count and degrees, Euler's vertex-face-edge equation, total boundary-incidence count, and the lower bound of three on every face degree.
Definition code
import Mathlib
set_option autoImplicit false
structure PlanarMapData__p2m_b23772a1_7f55_4be6_b745_ebb5a4e685d3_a3
{V : Type} [Fintype V]
(J : SimpleGraph V) [DecidableRel J.Adj] where
faceCount : Nat
euler : Fintype.card V + faceCount = J.edgeFinset.card + 2
faceDegree : Fin faceCount -> Nat
sum_faceDegree : Finset.sum Finset.univ faceDegree = 2 * J.edgeFinset.card
faceDegree_ge_three : forall f, 3 <= faceDegree fSource
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).