A topological sort is an ordering of the vertices of a
*directed acyclic graph*
given by the following definition:

Definition (Topological Sort)Consider a directed acyclic graph . Atopological sortof the vertices ofGis a sequence in which each element of appears exactly once. For every pair of distinct vertices and in the sequenceS, if is an edge inG, i.e., , theni<j.

Informally, a topological sort is a list of the vertices of a DAG in which all the successors of any given vertex appear in the sequence after that vertex. Consider the directed acyclic graph shown in Figure . The sequence is a topological sort of the vertices of . To see that this is so, consider the set of vertices:

The vertices in each edge are in alphabetical order,
and so is the sequence *S*.

**Figure:** A Directed Acyclic Graph

It should also be evident from Figure that a topological sort is not unique. For example, the following are also valid topological sorts of the graph :

One way to find a topological sort
is to consider the *in-degrees* of the vertices.
(The number above a vertex in Figure is the in-degree of that vertex).
Clearly the first vertex in a topological sort must have in-degree zero and
every DAG must contain at least one vertex with in-degree zero.
A simple algorithm to create the sort goes like this:

Repeat the following steps until the graph is empty:

- Select a vertex that has in-degree zero.
- Add the vertex to the sort.
- Delete the vertex and all the edges emanating from it from the graph.

Copyright © 1997 by Bruno R. Preiss, P.Eng. All rights reserved.