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Convert undirected connected graph to strongly connected directed graph
• Difficulty Level : Hard
• Last Updated : 15 Dec, 2020

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++

 `// C++ program for the above approach` `#include ``using` `namespace` `std;` `// To store the assigned Edges``vector > 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[],``                 ``vector<``int``> adj[],``                 ``int` `M)``{` `    ``// Traverse the Edges``    ``for` `(``int` `i = 0; i < M; i++) {` `        ``int` `u = Edges[i];``        ``int` `v = Edges[i];` `        ``// 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]``        ``= { { 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;``}`

## Java

 `// 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> 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> 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> 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> 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`

## Python3

 `# 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`

## C#

 `// C# program for the above approach``using` `System;``using` `System.Collections;``using` `System.Collections.Generic;`` ` `class` `GFG{``     ` `// To store the assigned Edges``static` `ArrayList ans;``  ` `// Flag variable to check Bridges``static` `int` `flag = 1;``  ` `// Function to implement DFS Traversal``static` `int` `dfs(ArrayList adj,``               ``int``[] order, ``int``[] bridge_detect,``               ``bool``[] 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 < ((ArrayList)adj[v]).Count;``            ``i++)``    ``{``        ``int` `u = (``int``)((ArrayList)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] = 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 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];``    ``bool` `[]mark = ``new` `bool``[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)``    ``{``        ``Console.Write(``"-1"``);``    ``}``    ` `    ``// Else print the direction of``    ``// Edges assigned``    ``else``    ``{``        ``foreach``(``int``[] it ``in` `ans)``        ``{``            ``Console.WriteLine(it + ``"->"` `+``                              ``it);``        ``}``    ``}``}``  ` `// Function to create graph``static` `void` `createGraph(``int` `[,]Edges,``                        ``ArrayList 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``        ``((ArrayList)adj[u]).Add(v);``        ``((ArrayList)adj[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 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 pratham76`
Output:
```0->1
2->0
4->1
3->4
2->3
1->2```

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