Convert undirected connected graph to strongly connected directed graph

Given an undirected graph of N vertices and M edges, the task is to assign directions to the given M Edges such that the graph becomes Strongly Connected Components. If a graph cannot be converted into Strongly Connected Components then print “-1”.
Examples: 

Input: N = 5, Edges[][] = { { 0, 1 }, { 0, 2 }, { 1, 2 }, { 1, 4 }, { 2, 3 }, { 3, 4 } } 
Output: 
0->1 
2->0 
4->1 
3->4 
2->3 
1->2 
Explanation: 
Below is the assigned edges to the above undirected graph: 
 

Input: N = 5, Edges[][] = { { 0, 1 }, { 0, 2 }, { 1, 3 }, { 2, 3 }, { 3, 4 } } 
Output: -1 
Explanation: 
Below is the graph for the above information: 
 



Since there is a bridge present in the above-undirected graph. Therefore, this graph can’t be converted into SCCs. 

Approach: We know that in any directed graph is said to be in Strongly Connected Components(SCCs) iff all the vertices of the graph are a part of some cycle. The given undirected graph doesn’t form SCCs if and only if the graph contains any bridges in it. Below are the steps: 

  • We will use an array mark[] to store the visited node during DFS Traversal, order[] to store the index number of the visited node, and bridge_detect[] to store any bridge present in the given graph.
  • Start the DFS Traversal from vertex 1.
  • Traverse the Adjacency list of current Node and do the following: 
    • If any edges are traverse again while any DFS call then ignore that edges.
    • If the order of child Node(Node u) is greater than the order of parent node(node v), then ignore this current edges as as Edges(v, u) is already processed.
    • If any Back Edge is found then update the Bridge Edges of the current parent node(node v) as:
 bridge_detect[v] = min(order[u], bridge_detect[v]); 



  • Else do the DFS Traversal for the current child node and repeat step 3 for the current node.
  • Update the bridges detect after DFS call for the current node as:
bridge_detect[v] = min(bridge_detect[u], bridge_detect[v])



  • Store the current pair of Edges(v, u) as directed Edges from Node v to Node u in an array of pairs(say arr[][]).
  • If there is any bridge present in the given graph then print “-1”.
  • Else print the directed Edges stored in arr[][].

Below is the implementation of the above approach: 

C++

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// C++ program for the above approach
 
#include <bits/stdc++.h>
using namespace std;
 
// To store the assigned Edges
vector<pair<int, int> > ans;
 
// Flag variable to check Bridges
int flag = 1;
 
// Function to implement DFS Traversal
int dfs(vector<int> adj[],
        int* order, int* bridge_detect,
        bool* mark, int v, int l)
{
 
    // Mark the current node as visited
    mark[v] = 1;
 
    // Update the order of node v
    order[v] = order[l] + 1;
 
    // Update the bridge_detect for node v
    bridge_detect[v] = order[v];
 
    // Traverse the adjacency list of
    // Node v
    for (int i = 0; i < adj[v].size(); i++) {
        int u = adj[v][i];
 
        // Ignores if same edge is traversed
        if (u == l) {
            continue;
        }
 
        // Ignores the edge u --> v as
        // v --> u is already processed
        if (order[v] < order[u]) {
            continue;
        }
 
        // Finds a back Edges, cycle present
        if (mark[u]) {
 
            // Update the bridge_detect[v]
            bridge_detect[v]
                = min(order[u],
                      bridge_detect[v]);
        }
 
        // Else DFS traversal for current
        // node in the adjacency list
        else {
 
            dfs(adj, order, bridge_detect,
                mark, u, v);
        }
 
        // Update the bridge_detect[v]
        bridge_detect[v]
            = min(bridge_detect[u],
                  bridge_detect[v]);
 
        // Store the current directed Edge
        ans.push_back(make_pair(v, u));
    }
 
    // Condition for Bridges
    if (bridge_detect[v] == order[v]
        && l != 0) {
        flag = 0;
    }
 
    // Return flag
    return flag;
}
 
// Function to print the direction
// of edges to make graph SCCs
void convert(vector<int> adj[], int n)
{
 
    // Arrays to store the visited,
    // bridge_detect and order of
    // Nodes
    int order[n] = { 0 };
    int bridge_detect[n] = { 0 };
    bool mark[n];
 
    // Intialise marks[] as false
    memset(mark, false, sizeof(mark));
 
    // DFS Traversal from vertex 1
    int flag = dfs(adj, order,
                   bridge_detect,
                   mark, 1, 0);
 
    // If flag is zero, then Bridge
    // is present in the graph
    if (flag == 0) {
        cout << "-1";
    }
 
    // Else print the direction of
    // Edges assigned
    else {
        for (auto& it : ans) {
            cout << it.first << "->"
                 << it.second << '\n';
        }
    }
}
 
// Function to create graph
void createGraph(int Edges[][2],
                 vector<int> adj[],
                 int M)
{
 
    // Traverse the Edges
    for (int i = 0; i < M; i++) {
 
        int u = Edges[i][0];
        int v = Edges[i][1];
 
        // Push the edges in an
        // adjacency list
        adj[u].push_back(v);
        adj[v].push_back(u);
    }
}
 
// Driver Code
int main()
{
    // N vertices and M Edges
    int N = 5, M = 6;
    int Edges[M][2]
        = { { 0, 1 }, { 0, 2 },
            { 1, 2 }, { 1, 4 },
            { 2, 3 }, { 3, 4 } };
 
    // To create Adjacency List
    vector<int> adj[N];
 
    // Create an undirected graph
    createGraph(Edges, adj, M);
 
    // Function Call
    convert(adj, N);
    return 0;
}

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Java

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// Java program for the above approach
import java.util.*;
import java.lang.*;
 
class GFG{
     
// To store the assigned Edges
static ArrayList<int[]> ans;
  
// Flag variable to check Bridges
static int flag = 1;
  
// Function to implement DFS Traversal
static int dfs(ArrayList<ArrayList<Integer>> adj,
               int[] order, int[] bridge_detect,
               boolean[] mark, int v, int l)
{
     
    // Mark the current node as visited
    mark[v] = true;
     
    // Update the order of node v
    order[v] = order[l] + 1;
  
    // Update the bridge_detect for node v
    bridge_detect[v] = order[v];
  
    // Traverse the adjacency list of
    // Node v
    for(int i = 0; i < adj.get(v).size(); i++)
    {
        int u = adj.get(v).get(i);
  
        // Ignores if same edge is traversed
        if (u == l)
        {
            continue;
        }
  
        // Ignores the edge u --> v as
        // v --> u is already processed
        if (order[v] < order[u])
        {
            continue;
        }
  
        // Finds a back Edges, cycle present
        if (mark[u])
        {
             
            // Update the bridge_detect[v]
            bridge_detect[v] = Math.min(order[u],
                                bridge_detect[v]);
        }
  
        // Else DFS traversal for current
        // node in the adjacency list
        else
        {
            dfs(adj, order, bridge_detect,
                mark, u, v);
        }
  
        // Update the bridge_detect[v]
        bridge_detect[v] = Math.min(bridge_detect[u],
                                    bridge_detect[v]);
  
        // Store the current directed Edge
        ans.add(new int[]{v, u});
    }
  
    // Condition for Bridges
    if (bridge_detect[v] == order[v] && l != 0)
    {
        flag = 0;
    }
     
    // Return flag
    return flag;
}
  
// Function to print the direction
// of edges to make graph SCCs
static void convert(ArrayList<ArrayList<Integer>> adj,
                    int n)
{
     
    // Arrays to store the visited,
    // bridge_detect and order of
    // Nodes
    int[] order = new int[n];
    int[] bridge_detect = new int[n];
    boolean mark[] = new boolean[n];
     
    // DFS Traversal from vertex 1
    int flag = dfs(adj, order,
                   bridge_detect,
                   mark, 1, 0);
  
    // If flag is zero, then Bridge
    // is present in the graph
    if (flag == 0)
    {
        System.out.print("-1");
    }
  
    // Else print the direction of
    // Edges assigned
    else
    {
        for(int[] it : ans)
        {
            System.out.println(it[0] + "->" +
                               it[1]);
        }
    }
}
  
// Function to create graph
static void createGraph(int Edges[][],
                        ArrayList<ArrayList<Integer>> adj,
                        int M)
{
     
    // Traverse the Edges
    for(int i = 0; i < M; i++)
    {
        int u = Edges[i][0];
        int v = Edges[i][1];
         
        // Push the edges in an
        // adjacency list
        adj.get(u).add(v);
        adj.get(v).add(u);
    }
}
 
// Driver code
public static void main(String[] args)
{
     
    // N vertices and M Edges
    int N = 5, M = 6;
     
    int Edges[][] = { { 0, 1 }, { 0, 2 },
                      { 1, 2 }, { 1, 4 },
                      { 2, 3 }, { 3, 4 } };
     
    // To create Adjacency List
    ArrayList<ArrayList<Integer>> adj = new ArrayList<>();
    ans = new ArrayList<>();
     
    for(int i = 0; i < N; i++)
        adj.add(new ArrayList<>());
     
    // Create an undirected graph
    createGraph(Edges, adj, M);
     
    // Function Call
    convert(adj, N);
}
}
 
// This code is contributed by offbeat

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Python3

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# Python3 program for
# the above approach
 
# To store the assigned
# Edges
ans = []
  
# Flag variable to
# check Bridges
flag = 1;
  
# Function to implement
# DFS Traversal
def dfs(adj, order,
        bridge_detect,
        mark, v, l):
     
    global flag
     
    # Mark the current
    # node as visited
    mark[v] = 1;
  
    # Update the order of
    # node v
    order[v] = order[l] + 1;
  
    # Update the bridge_detect
    # for node v
    bridge_detect[v] = order[v];
  
    # Traverse the adjacency list of
    # Node v
    for i in range(len(adj[v])):       
        u = adj[v][i];
  
        # Ignores if same edge
        # is traversed
        if (u == l):
            continue;      
  
        # Ignores the edge u --> v as
        # v --> u is already processed
        if (order[v] < order[u]):
            continue;       
  
        # Finds a back Edges,
        # cycle present
        if (mark[u]):
  
            # Update the bridge_detect[v]
            bridge_detect[v] = min(order[u],
                                  bridge_detect[v]);
         
        # Else DFS traversal for current
        # node in the adjacency list
        else:
  
            dfs(adj, order,
                bridge_detect,
                mark, u, v);       
  
        # Update the bridge_detect[v]
        bridge_detect[v] = min(bridge_detect[u],
                              bridge_detect[v]);
  
        # Store the current
        # directed Edge
        ans.append([v, u]);
  
    # Condition for Bridges
    if (bridge_detect[v] ==
        order[v] and l != 0):
        flag = 0;
     
    # Return flag
    return flag;
  
# Function to print the
# direction of edges to
# make graph SCCs
def convert(adj, n):
  
    # Arrays to store the visited,
    # bridge_detect and order of
    # Nodes
    order = [0 for i in range(n)]
    bridge_detect = [0 for i in range(n)]
    mark = [False for i in range(n)]
  
    # DFS Traversal from
    # vertex 1
    flag = dfs(adj, order,
               bridge_detect,
               mark, 1, 0);
  
    # If flag is zero, then Bridge
    # is present in the graph
    if (flag == 0):
        print(-1)
  
    # Else print the direction
    # of Edges assigned
    else:
        for it in ans:
            print("{} -> {}".format(it[0],
                                    it[1]))
 
# Function to create graph
def createGraph(Edges,adj, M):
  
    # Traverse the Edges
    for i in range(M):
  
        u = Edges[i][0];
        v = Edges[i][1];
  
        # Push the edges in an
        # adjacency list
        adj[u].append(v);
        adj[v].append(u);
 
# Driver code
if __name__ == "__main__":
     
    # N vertices and M Edges
    N = 5
    M = 6;
    Edges = [[0, 1], [0, 2],
            [1, 2], [1, 4],
            [2, 3], [3, 4]];
  
    # To create Adjacency List
    adj = [[] for i in range(N)]
  
    # Create an undirected graph
    createGraph(Edges, adj, M);
  
    # Function Call
    convert(adj, N);
 
# This code is contributed by rutvik_56

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Output: 

0->1
2->0
4->1
3->4
2->3
1->2




 

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