Data Modelling & AI Data Structure & Algorithm

All Topological Sorts of a Directed Acyclic Graph

21 June 2025

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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.

Given a DAG, print all topological sorts of the graph.

For example, consider the below graph.

graph

All topological sorts of the given graph are:
4 5 0 2 3 1 
4 5 2 0 3 1 
4 5 2 3 0 1 
4 5 2 3 1 0 
5 2 3 4 0 1 
5 2 3 4 1 0 
5 2 4 0 3 1 
5 2 4 3 0 1 
5 2 4 3 1 0 
5 4 0 2 3 1 
5 4 2 0 3 1 
5 4 2 3 0 1 
5 4 2 3 1 0

In a Directed acyclic graph many a times we can have vertices which are unrelated to each other because of which we can order them in many ways. These various topological sorting is important in many cases, for example if some relative weight is also available between the vertices, which is to minimize then we need to take care of relative ordering as well as their relative weight, which creates the need of checking through all possible topological ordering.

We can go through all possible ordering via backtracking , the algorithm step are as follows :

Initialize all vertices as unvisited.

Now choose vertex which is unvisited and has zero indegree and decrease indegree of all those vertices by 1 (corresponding to removing edges) now add this vertex to result and call the recursive function again and backtrack.

After returning from function reset values of visited, result and indegree for enumeration of other possibilities.

Below is the implementation of the above steps.

C++

// C++ program to print all topological sorts of a graph
#include <bits/stdc++.h>
using namespace std;
 
class Graph
{
    int V;    // No. of vertices
 
    // Pointer to an array containing adjacency list
    list<int> *adj;
 
    // Vector to store indegree of vertices
    vector<int> indegree;
 
    // A function used by alltopologicalSort
    void alltopologicalSortUtil(vector<int>& res,
                                bool visited[]);
 
public:
    Graph(int V);   // Constructor
 
    // function to add an edge to graph
    void addEdge(int v, int w);
 
    // Prints all Topological Sorts
    void alltopologicalSort();
};
 
//  Constructor of graph
Graph::Graph(int V)
{
    this->V = V;
    adj = new list<int>[V];
 
    // Initialising all indegree with 0
    for (int i = 0; i < V; i++)
        indegree.push_back(0);
}
 
//  Utility function to add edge
void Graph::addEdge(int v, int w)
{
    adj[v].push_back(w); // Add w to v's list.
 
    // increasing inner degree of w by 1
    indegree[w]++;
}
 
//  Main recursive function to print all possible
//  topological sorts
void Graph::alltopologicalSortUtil(vector<int>& res,
                                   bool visited[])
{
    // To indicate whether all topological are found
    // or not
    bool flag = false; 
 
    for (int i = 0; i < V; i++)
    {
        //  If indegree is 0 and not yet visited then
        //  only choose that vertex
        if (indegree[i] == 0 && !visited[i])
        {
            //  reducing indegree of adjacent vertices
            list<int>:: iterator j;
            for (j = adj[i].begin(); j != adj[i].end(); j++)
                indegree[*j]--;
 
            //  including in result
            res.push_back(i);
            visited[i] = true;
            alltopologicalSortUtil(res, visited);
 
            // resetting visited, res and indegree for
            // backtracking
            visited[i] = false;
            res.erase(res.end() - 1);
            for (j = adj[i].begin(); j != adj[i].end(); j++)
                indegree[*j]++;
 
            flag = true;
        }
    }
 
    //  We reach here if all vertices are visited.
    //  So we print the solution here
    if (!flag)
    {
        for (int i = 0; i < res.size(); i++)
            cout << res[i] << " ";
        cout << endl;
    }
}
 
//  The function does all Topological Sort.
//  It uses recursive alltopologicalSortUtil()
void Graph::alltopologicalSort()
{
    // Mark all the vertices as not visited
    bool *visited = new bool[V];
    for (int i = 0; i < V; i++)
        visited[i] = false;
 
    vector<int> res;
    alltopologicalSortUtil(res, visited);
}
 
// 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 << "All Topological sorts\n";
 
    g.alltopologicalSort();
 
    return 0;
}

Java

//Java program to print all topological sorts of a graph
import java.util.*;
 
class Graph {
    int V; // No. of vertices
 
    List<Integer> adjListArray[];
 
    public Graph(int V) {
 
        this.V = V;
 
        @SuppressWarnings("unchecked")
        List<Integer> adjListArray[] = new LinkedList[V];
 
        this.adjListArray = adjListArray;
 
        for (int i = 0; i < V; i++) {
            adjListArray[i] = new LinkedList<>();
        }
    }
    // Utility function to add edge
    public void addEdge(int src, int dest) {
 
        this.adjListArray[src].add(dest);
 
    }
     
    // Main recursive function to print all possible
    // topological sorts
    private void allTopologicalSortsUtil(boolean[] visited, 
                        int[] indegree, ArrayList<Integer> stack) {
        // To indicate whether all topological are found
        // or not
        boolean flag = false;
 
        for (int i = 0; i < this.V; i++) {
            // If indegree is 0 and not yet visited then
            // only choose that vertex
            if (!visited[i] && indegree[i] == 0) {
                 
                // including in result
                visited[i] = true;
                stack.add(i);
                for (int adjacent : this.adjListArray[i]) {
                    indegree[adjacent]--;
                }
                allTopologicalSortsUtil(visited, indegree, stack);
                 
                // resetting visited, res and indegree for
                // backtracking
                visited[i] = false;
                stack.remove(stack.size() - 1);
                for (int adjacent : this.adjListArray[i]) {
                    indegree[adjacent]++;
                }
 
                flag = true;
            }
        }
        // We reach here if all vertices are visited.
        // So we print the solution here
        if (!flag) {
            stack.forEach(i -> System.out.print(i + " "));
            System.out.println();
        }
 
    }
     
    // The function does all Topological Sort.
    // It uses recursive alltopologicalSortUtil()
    public void allTopologicalSorts() {
        // Mark all the vertices as not visited
        boolean[] visited = new boolean[this.V];
 
        int[] indegree = new int[this.V];
 
        for (int i = 0; i < this.V; i++) {
 
            for (int var : this.adjListArray[i]) {
                indegree[var]++;
            }
        }
 
        ArrayList<Integer> stack = new ArrayList<>();
 
        allTopologicalSortsUtil(visited, indegree, stack);
    }
     
    // Driver code
    public static void main(String[] args) {
 
        // Create a graph given in the above diagram
        Graph graph = new Graph(6);
        graph.addEdge(5, 2);
        graph.addEdge(5, 0);
        graph.addEdge(4, 0);
        graph.addEdge(4, 1);
        graph.addEdge(2, 3);
        graph.addEdge(3, 1);
 
        System.out.println("All Topological sorts");
        graph.allTopologicalSorts();
    }
}

Python3

# class to represent a graph object
class Graph:
 
    # Constructor
    def __init__(self, edges, N):
 
        # A List of Lists to represent an adjacency list
        self.adjList = [[] for _ in range(N)]
 
        # stores in-degree of a vertex
        # initialize in-degree of each vertex by 0
        self.indegree = [0] * N
 
        # add edges to the undirected graph
        for (src, dest) in edges:
 
            # add an edge from source to destination
            self.adjList[src].append(dest)
 
            # increment in-degree of destination vertex by 1
            self.indegree[dest] = self.indegree[dest] + 1
 
 
# Recursive function to find 
# all topological orderings of a given DAG
def findAllTopologicalOrders(graph, path, discovered, N):
 
    # do for every vertex
    for v in range(N):
 
        # proceed only if in-degree of current node is 0 and
        # current node is not processed yet
        if graph.indegree[v] == 0 and not discovered[v]:
 
            # for every adjacent vertex u of v, 
            # reduce in-degree of u by 1
            for u in graph.adjList[v]:
                graph.indegree[u] = graph.indegree[u] - 1
 
            # include current node in the path 
            # and mark it as discovered
            path.append(v)
            discovered[v] = True
 
            # recur
            findAllTopologicalOrders(graph, path, discovered, N)
 
            # backtrack: reset in-degree 
            # information for the current node
            for u in graph.adjList[v]:
                graph.indegree[u] = graph.indegree[u] + 1
 
            # backtrack: remove current node from the path and
            # mark it as undiscovered
            path.pop()
            discovered[v] = False
 
    # print the topological order if 
    # all vertices are included in the path
    if len(path) == N:
        print(path)
 
 
# Print all topological orderings of a given DAG
def printAllTopologicalOrders(graph):
 
    # get number of nodes in the graph
    N = len(graph.adjList)
 
    # create an auxiliary space to keep track of whether vertex is discovered
    discovered = [False] * N
 
    # list to store the topological order
    path = []
 
    # find all topological ordering and print them
    findAllTopologicalOrders(graph, path, discovered, N)
 
# Driver code
if __name__ == '__main__':
 
    # List of graph edges as per above diagram
    edges = [(5, 2), (5, 0), (4, 0), (4, 1), (2, 3), (3, 1)]
 
    print("All Topological sorts")
 
    # Number of nodes in the graph
    N = 6
 
    # create a graph from edges
    graph = Graph(edges, N)
 
    # print all topological ordering of the graph
    printAllTopologicalOrders(graph)
 
# This code is contributed by Priyadarshini Kumari

C#

using System;
using System.Collections.Generic;
 
class Graph
{
    int V;
    List<int>[] adjListArray;
 
    public Graph(int V)
    {
        this.V = V;
        adjListArray = new List<int>[V];
 
        for (int i = 0; i < V; i++)
        {
            adjListArray[i] = new List<int>();
        }
    }
 
    public void addEdge(int src, int dest)
    {
        this.adjListArray[src].Add(dest);
    }
 
    private void allTopologicalSortsUtil(bool[] visited, int[] indegree, List<int> stack)
    {
        bool flag = false;
 
        for (int i = 0; i < this.V; i++)
        {
            if (!visited[i] && indegree[i] == 0)
            {
                visited[i] = true;
                stack.Add(i);
                foreach (int adjacent in this.adjListArray[i])
                {
                    indegree[adjacent]--;
                }
                allTopologicalSortsUtil(visited, indegree, stack);
 
                visited[i] = false;
                stack.RemoveAt(stack.Count - 1);
                foreach (int adjacent in this.adjListArray[i])
                {
                    indegree[adjacent]++;
                }
 
                flag = true;
            }
        }
        if (!flag)
        {
            stack.ForEach(i => Console.Write(i + " "));
            Console.WriteLine();
        }
    }
 
    public void allTopologicalSorts()
    {
        bool[] visited = new bool[this.V];
        int[] indegree = new int[this.V];
 
        for (int i = 0; i < this.V; i++)
        {
            foreach (int var in this.adjListArray[i])
            {
                indegree[var]++;
            }
        }
 
        List<int> stack = new List<int>();
 
        allTopologicalSortsUtil(visited, indegree, stack);
    }
 
    static void Main(string[] args)
    {
        Graph graph = new Graph(6);
        graph.addEdge(5, 2);
        graph.addEdge(5, 0);
        graph.addEdge(4, 0);
        graph.addEdge(4, 1);
        graph.addEdge(2, 3);
        graph.addEdge(3, 1);
 
        Console.WriteLine("All Topological sorts");
        graph.allTopologicalSorts();
    }
}

Javascript

<script>
 
// class to represent a graph object
class Graph{
    // Constructor 
    constructor(edges, N){
         
        // A List of Lists to represent an adjacency list
        this.adjList = new Array(N);
        for(let i = 0; i < N; i++){
            this.adjList[i] = new Array();
        }
         
        // stores in-degree of a vertex
        // initialize in-degree of each vertex by 0
        this.indegree = new Array(N).fill(0);
         
        // add edges to the undirected graph
        for(let i = 0; i < edges.length; i++){
            let src = edges[i][0];
            let dest = edges[i][1];
             
            //add an edge from source to destination
            this.adjList[src].push(dest);
             
            // increment in-degree of destination vertex by 1
            this.indegree[dest] = this.indegree[dest] + 1;
        }
             
    }
}
 
 
// Recursive function to find 
// all topological orderings of a given DAG
function findAllTopologicalOrders(graph, path, discovered, N){
  
    // do for every vertex
    for(let v = 0; v < N; v++){
         
        // proceed only if in-degree of current node is 0 and
        // current node is not processed yet
        if(graph.indegree[v] == 0 &&  !discovered[v]){
             
            // for every adjacent vertex u of v, 
            // reduce in-degree of u by 1
            for(let indx = 0; indx < graph.adjList[v].length; indx++){
                let u = graph.adjList[v][indx];
                graph.indegree[u] = graph.indegree[u] - 1;
            }
        }
         
        // include current node in the path 
        // and mark it as discovered
        path.push(v);
        discovered[v] = true;
         
        // recur
        findAllTopologicalOrders(graph, path, discovered, N)
 
        // backtrack: reset in-degree 
        // information for the current node
        for(let indx = 0; indx < graph.adjList[v].length; indx++){
            let u = graph.adjList[v][indx];
            graph.indegree[u] = graph.indegree[u] + 1;
        }
 
        // backtrack: remove current node from the path and
        // mark it as undiscovered
        path.pop();
        discovered[v] = false;
 
    }
     
     
    // print the topological order if 
    // all vertices are included in the path
    if(path.length == N){
        console.log(path);
    }
}
 
     
 
 
// Print all topological orderings of a given DAG
function printAllTopologicalOrders(graph){
     
    // get number of nodes in the graph
    let N = graph.adjList.length;
     
    // create an auxiliary space to keep track of whether vertex is discovered
    let discovered = new Array(N).fill(false);
     
    // list to store the topological order
    let path = [];
     
    // find all topological ordering and print them
    findAllTopologicalOrders(graph, path, discovered, N)
}
 
 
// Driver code
 
// List of graph edges as per above diagram
let edges = [[5, 2], [5, 0], [4, 0], [4, 1], [2, 3], [3, 1]];
 
console.log("All Topological sorts");
 
// Number of nodes in the graph
let N = 6;
 
// create a graph from edges
let graph = new Graph(edges, N);
 
// print all topological ordering of the graph
printAllTopologicalOrders(graph);
 
// This code is contributed by gautam goel. 
 
 
 
</script>

Output

All Topological sorts
4 5 0 2 3 1 
4 5 2 0 3 1 
4 5 2 3 0 1 
4 5 2 3 1 0 
5 2 3 4 0 1 
5 2 3 4 1 0 
5 2 4 0 3 1 
5 2 4 3 0 1 
5 2 4 3 1 0 
5 4 0 2 3 1 
5 4 2 0 3 1 
5 4 2 3 0 1 
5 4 2 3 1 0

Time Complexity: O(V!), Here V is the number of vertices, V! is absolute worst case. (worst case example – any graph with no edges at all)
Auxiliary Space: O(V), for creating an additional array and recursive stack space.

This articles is contributed by Utkarsh Trivedi. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

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All Topological Sorts of a Directed Acyclic Graph

C++

Java

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Javascript

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