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Lax195003.WelzlOrdersNeighborhoodSetSystem

Neighborhood set systems of graphs

concepts/Lax195003/WelzlOrdersNeighborhoodSetSystem.lean · lax-195003

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    Definition

    For a graph and a radius k, its neighborhood set system consists of the open k-neighborhood of every vertex. The open k-neighborhood of a vertex contains every other vertex reachable from it by a walk of length at most k.

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    1import Lax195003.WelzlOrders
    2import Mathlib.Combinatorics.SimpleGraph.Walk.Basic
    3
    4/-!
    5---
    6title: Neighborhood set systems of graphs
    7type: definition
    8---
    9For a graph and a radius *k*, its neighborhood set system consists of the
    10open *k*-neighborhood of every vertex. The open *k*-neighborhood of a vertex
    11contains every other vertex reachable from it by a walk of length at most
    12*k*.
    13
    14# Formalization notes
    15
    16The center vertex is excluded explicitly, so the radius-one instance is the
    17ordinary open neighborhood set system of a simple graph.
    18-/
    19
    20namespace Lax195003.WelzlOrdersNeighborhoodSetSystem
    21
    22open Lax195003.WelzlOrders
    23
    24/-- The neighborhood set system of a graph at radius `k`: the open
    25`k`-neighborhood of every vertex. -/
    26def neighborhoodSetSystem {n : ℕ} (G : SimpleGraph (Fin n)) (k : ℕ) :
    27 SetSystem (Fin n) :=
    28 {X | ∃ v : Fin n, X =
    29 {u : Fin n | u ≠ v ∧ ∃ w : G.Walk v u, w.length ≤ k}}
    30
    31end Lax195003.WelzlOrdersNeighborhoodSetSystem
    32

    Formalization notes

    The center vertex is excluded explicitly, so the radius-one instance is the ordinary open neighborhood set system of a simple graph.

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