Topological Sort of a graph using departure time of vertex
Last Updated :
23 Jul, 2025
Given a Directed Acyclic Graph (DAG), find Topological Sort of the graph.
Topological sorting for Directed Acyclic Graph (DAG) is a linear ordering of vertices such that for every directed edge uv, vertex u comes before v in the ordering. Topological Sorting for a graph is not possible if the graph is not a DAG.
For example, a topological sorting of the following graph is "5 4 2 3 1 0". There can be more than one topological sorting for a graph. For example, another topological sorting of the following graph is "4 5 2 3 1 0".
Please note that the first vertex in topological sorting is always a vertex with in-degree as 0 (a vertex with no incoming edges). For above graph, vertex 4 and 5 have no incoming edges.
We have already discussed a DFS-based algorithm using stack and Kahn’s Algorithm for Topological Sorting. We have also discussed how to print all topological sorts of the DAG here. In this post, another DFS based approach is discussed for finding Topological sort of a graph by introducing concept of arrival and departure time of a vertex in DFS.
What is Arrival Time & Departure Time of Vertices in DFS?
In DFS, Arrival Time is the time at which the vertex was explored for the first time and Departure Time is the time at which we have explored all the neighbors of the vertex and we are ready to backtrack.
How to find Topological Sort of a graph using departure time?
To find Topological Sort of a graph, we run DFS starting from all unvisited vertices one by one. For any vertex, before exploring any of its neighbors, we note the arrival time of that vertex and after exploring all the neighbors of the vertex, we note its departure time. Please note only departure time is needed to find Topological Sort of a graph, so we can skip arrival time of vertex. Finally, after we have visited all the vertices of the graph, we print the vertices in order of their decreasing departure time which is our desired Topological Order of Vertices.
Below is C++ implementation of above idea –
C++
// A C++ program to print topological sorting of a DAG
#include <bits/stdc++.h>
using namespace std;
// Graph class represents a directed graph using adjacency
// list representation
class Graph {
int V; // No. of vertices
// Pointer to an array containing adjacency lists
list<int>* adj;
public:
Graph(int); // Constructor
~Graph(); // Destructor
// function to add an edge to graph
void addEdge(int, int);
// The function to do DFS traversal
void DFS(int, vector<bool>&, vector<int>&, int&);
// The function to do Topological Sort.
void topologicalSort();
};
Graph::Graph(int V)
{
this->V = V;
this->adj = new list<int>[V];
}
Graph::~Graph() { delete[] this->adj; }
void Graph::addEdge(int v, int w)
{
adj[v].push_back(w); // Add w to v's list.
}
// The function to do DFS() and stores departure time
// of all vertex
void Graph::DFS(int v, vector<bool>& visited,
vector<int>& departure, int& time)
{
visited[v] = true;
// time++; // arrival time of vertex v
for (int i : adj[v])
if (!visited[i])
DFS(i, visited, departure, time);
// set departure time of vertex v
departure[time++] = v;
}
// The function to do Topological Sort. It uses DFS().
void Graph::topologicalSort()
{
// vector to store departure time of vertex.
vector<int> departure(V, -1);
// Mark all the vertices as not visited
vector<bool> visited(V, false);
int time = 0;
// perform DFS on all unvisited vertices
for (int i = 0; i < V; i++) {
if (visited[i] == 0) {
DFS(i, visited, departure, time);
}
}
// print the topological sort
for (int i = V - 1; i >= 0; i--)
cout << departure[i] << " ";
}
// Driver program to test above functions
int main()
{
// Create a graph given in the above diagram
Graph g(6);
g.addEdge(5, 2);
g.addEdge(5, 0);
g.addEdge(4, 0);
g.addEdge(4, 1);
g.addEdge(2, 3);
g.addEdge(3, 1);
cout << "Topological Sort of the given graph is \n";
g.topologicalSort();
return 0;
}
# A Python3 program to print topological sorting of a DAG
def addEdge(u, v):
global adj
adj[u].append(v)
# The function to do DFS() and stores departure time
# of all vertex
def DFS(v):
global visited, departure, time
visited[v] = 1
for i in adj[v]:
if visited[i] == 0:
DFS(i)
departure[time] = v
time += 1
# The function to do Topological Sort. It uses DFS().
def topologicalSort():
# perform DFS on all unvisited vertices
for i in range(V):
if(visited[i] == 0):
DFS(i)
# Print vertices in topological order
for i in range(V - 1, -1, -1):
print(departure[i], end = " ")
# Driver code
if __name__ == '__main__':
# Create a graph given in the above diagram
V,time, adj, visited, departure = 6, 0, [[] for i in range(7)], [0 for i in range(7)],[-1 for i in range(7)]
addEdge(5, 2)
addEdge(5, 0)
addEdge(4, 0)
addEdge(4, 1)
addEdge(2, 3)
addEdge(3, 1)
print("Topological Sort of the given graph is")
topologicalSort()
# This code is contributed by mohit kumar 29
// C# program to print topological sorting of a DAG
using System;
using System.Collections.Generic;
// Graph class represents a directed graph using adjacency
// list representation
public class Graph {
private int V;
private List<int>[] adj;
// constructor
public Graph(int v)
{
V = v;
adj = new List<int>[ v ];
for (int i = 0; i < v; i++)
adj[i] = new List<int>();
}
// Add an edge
public void AddEdge(int v, int w)
{
adj[v].Add(w); // Add w to v's list
}
// The function to do DFS() and stores departure time
// of all vertex
private void DFS(int v, bool[] visited, int[] departure,
ref int time)
{
visited[v] = true;
// time++; // arrival time of vertex v
foreach(int i in adj[v])
{
if (!visited[i])
DFS(i, visited, departure, ref time);
}
// set departure time of vertex v
departure[time++] = v;
}
// The function to do Topological Sort. It uses DFS().
public void TopologicalSort()
{
// vector to store departure time of vertex.
int[] departure = new int[V];
for (int i = 0; i < V; i++)
departure[i] = -1;
// Mark all the vertices as not visited
bool[] visited = new bool[V];
int time = 0;
// perform DFS on all unvisited vertices
for (int i = 0; i < V; i++) {
if (visited[i] == false) {
DFS(i, visited, departure, ref time);
}
}
// print the topological sort
for (int i = V - 1; i >= 0; i--)
Console.Write(departure[i] + " ");
Console.WriteLine();
}
}
class GFG {
// Driver program to test above functions
static void Main(string[] args)
{
// Create a graph given in the above diagram
Graph g = new Graph(6);
g.AddEdge(5, 2);
g.AddEdge(5, 0);
g.AddEdge(4, 0);
g.AddEdge(4, 1);
g.AddEdge(2, 3);
g.AddEdge(3, 1);
Console.WriteLine(
"Topological Sort of the given graph is");
g.TopologicalSort();
}
}
// This code is contributed by cavi4762
<script>
// A JavaScript program to print topological
// sorting of a DAG
let adj=new Array(7);
for(let i=0;i<adj.length;i++)
{
adj[i]=[];
}
let V=6;
let time=0;
let visited=new Array(7);
let departure =new Array(7);
for(let i=0;i<7;i++)
{
visited[i]=0;
departure[i]=-1;
}
function addEdge(u, v)
{
adj[u].push(v)
}
// The function to do DFS() and
// stores departure time
// of all vertex
function DFS(v)
{
visited[v] = 1;
for(let i=0;i<adj[v].length;i++)
{
if(visited[adj[v][i]]==0)
DFS(adj[v][i]);
}
departure[time] = v
time += 1
}
// The function to do Topological
// Sort. It uses DFS().
function topologicalSort()
{
//perform DFS on all unvisited vertices
for(let i=0;i<V;i++)
{
if(visited[i] == 0)
DFS(i)
}
//perform DFS on all unvisited vertices
for(let i=V-1;i>=0;i--)
{
document.write(departure[i]+" ");
}
}
// Create a graph given in the above diagram
addEdge(5, 2);
addEdge(5, 0);
addEdge(4, 0);
addEdge(4, 1);
addEdge(2, 3);
addEdge(3, 1);
document.write(
"Topological Sort of the given graph is<br>"
);
topologicalSort()
// This code is contributed by unknown2108
</script>
import java.util.ArrayList;
// Graph class represents a directed graph using adjacency
// list representation
public class GFG {
int V;
ArrayList<ArrayList<Integer> > adj;
int time = 0;
// constructor
public GFG(int v)
{
V = v;
adj = new ArrayList<>();
for (int i = 0; i < v; i++)
adj.add(new ArrayList<>());
}
// Add an edge
public void AddEdge(int v, int w)
{
adj.get(v).add(w); // Add w to v's list
}
// The function to do DFS() and stores departure time
// of all vertex
private void DFS(int v, boolean[] visited,
int[] departure)
{
visited[v] = true;
// time++; // arrival time of vertex v
for (int i : adj.get(v)) {
if (!visited[i])
DFS(i, visited, departure);
}
// set departure time of vertex v
departure[time++] = v;
}
// The function to do Topological Sort. It uses DFS().
public void TopologicalSort()
{
// vector to store departure time of vertex.
int[] departure = new int[V];
for (int i = 0; i < V; i++)
departure[i] = -1;
// Mark all the vertices as not visited
boolean[] visited = new boolean[V];
// perform DFS on all unvisited vertices
for (int i = 0; i < V; i++) {
if (!visited[i]) {
DFS(i, visited, departure);
}
}
// print the topological sort
for (int i = V - 1; i >= 0; i--)
System.out.print(departure[i] + " ");
}
// Driver program to test above functions
public static void main(String[] args)
{
// Create a graph given in the above diagram
GFG g = new GFG(6);
g.AddEdge(5, 2);
g.AddEdge(5, 0);
g.AddEdge(4, 0);
g.AddEdge(4, 1);
g.AddEdge(2, 3);
g.AddEdge(3, 1);
System.out.println(
"Topological Sort of the given graph is");
g.TopologicalSort();
}
}
// This code is contributed by Prithi_Dey
OutputTopological Sort of the given graph is
5 4 2 3 1 0
Time Complexity of above solution is O(V + E).
Space Complexity of this algorithm is O(V). This is because we use a vector to store the departure time of each vertex, which is of size V.
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