# Definition:Arborescence/Definition 2

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## Definition

Let $G = \left({V, A}\right)$ be a directed graph.

Let $r \in V$.

$G$ is an **arborescence of root $r$** if and only if:

- $(1): \quad$ $G$ is an orientation of a tree
- $(2): \quad$ For each $v \in V$, $v$ is reachable from $r$.

### Root of Arborescence

The vertex $r$ of $G$ is known as the **root of (the arborescence) $G$**.

## Also known as

An arborescence of root $r$ can be referred to as an an **$r$-arborescence**, or just an **arborescence**.

Some sources, for example Donald E. Knuth: *The Art of Computer Programming: Volume 1: Fundamental Algorithms*, call an arborescence an **oriented tree**.

## Also defined as

Some sources, for example Donald E. Knuth: *The Art of Computer Programming: Volume 1: Fundamental Algorithms*, define an arborescence of root $r$ so as to reverse the orientation of $G$, so that the arcs are all directed toward the root rather than away from it.

## Also see

- Results about
**arborescences**can be found here.

## Sources

- May 2009: Kristóf Bérczi and András Frank:
*Packing Arborescences*(*Egerváry Research Group***Ser. Technical Report***no. TR-2009-04*): $\S 1$