# topologicalOrdering property

Set<T>? topologicalOrdering
inherited

Returns a set containing all graph vertices in topological order.

• For every directed edge: (vertex1 -> vertex2), vertex1 is listed before vertex2.

• Note: There is no topological ordering if the graph is cyclic. In that case the function returns `null`.

• Any self-loop (e.g. vertex1 -> vertex1) renders a directed graph cyclic.

• Based on a depth-first search algorithm (Cormen 2001, Tarjan 1976).

## Implementation

``````Set<T>? get topologicalOrdering {
final queue = Queue<T>();
final perm = HashSet<T>();
final temp = HashSet<T>();

// Marks graph as cyclic.
var isCyclic = false;

// Recursive function
void visit(T vertex) {
// Graph is not a Directed Acyclic Graph (DAG).
// Terminate iteration.
if (isCyclic) return;

// Vertex has permanent mark.
// => This vertex and its neighbouring vertices
// have already been visited.
if (perm.contains(vertex)) return;

// A cycle has been detected. Mark graph as acyclic.
if (temp.contains(vertex)) {
isCyclic = true;
return;
}

// Temporary mark. Marks current vertex as visited.
for (final connectedVertex in edges(vertex)) {
visit(connectedVertex);
}
// Permanent mark, indicating that there is no path from
// neighbouring vertices back to the current vertex.
// We tried all options.
}

// Main loop
// Note: Iterating in reverse order of [vertices]
// (approximately) preserves the
// sorting of vertices (on top of the topological sorting.)
// For a sorted topological ordering use
// the getter: [sortedTopologicalOrdering].
//
// Iterating in normal order of [vertices] yields a different
// valid topological sorting.
for (final vertex in sortedVertices.toList().reversed) {
visit(vertex);
if (isCyclic) break;
}

// Return null if graph is not a DAG.
return (isCyclic) ? null : queue.toSet();
}``````