Documentation

Mathlib.Combinatorics.Quiver.Arborescence

Arborescences #

A quiver V is an arborescence (or directed rooted tree) when we have a root vertex root : V such that for every b : V there is a unique path from root to b.

Main definitions #

class Quiver.Arborescence (V : Type u) [inst : Quiver V] :
Type (maxuv)
  • The root of the arborescence.

    root : V
  • There is a unique path from the root to any other vertex.

    uniquePath : (b : V) → Unique (Quiver.Path root b)

A quiver is an arborescence when there is a unique path from the default vertex to every other vertex.

Instances
    def Quiver.root (V : Type u) [inst : Quiver V] [inst : Quiver.Arborescence V] :
    V

    The root of an arborescence.

    Equations
    noncomputable def Quiver.arborescenceMk {V : Type u} [inst : Quiver V] (r : V) (height : V) (height_lt : ∀ ⦃a b : V⦄, (a b) → height a < height b) (unique_arrow : ∀ ⦃a b c : V⦄ (e : a c) (f : b c), a = b HEq e f) (root_or_arrow : ∀ (b : V), b = r a, Nonempty (a b)) :

    To show that [Quiver V] is an arborescence with root r : V, it suffices to

    • provide a height function V → ℕ→ ℕ such that every arrow goes from a lower vertex to a higher vertex,
    • show that every vertex has at most one arrow to it, and
    • show that every vertex other than r has an arrow to it.
    Equations
    • One or more equations did not get rendered due to their size.
    class Quiver.RootedConnected {V : Type u} [inst : Quiver V] (r : V) :

    RootedConnected r means that there is a path from r to any other vertex.

    Instances
      noncomputable def Quiver.shortestPath {V : Type u} [inst : Quiver V] (r : V) [inst : Quiver.RootedConnected r] (b : V) :

      A path from r of minimal length.

      Equations

      The length of a path is at least the length of the shortest path

      def Quiver.geodesicSubtree {V : Type u} [inst : Quiver V] (r : V) [inst : Quiver.RootedConnected r] :

      A subquiver which by construction is an arborescence.

      Equations
      Equations
      • One or more equations did not get rendered due to their size.