Given an undirected graph with V nodes (say numbered from 1 to V) and E edges, the task is to check whether the graph is an Euler Graph or not and if so then convert it into a Directed Euler Circuit.
A Directed Euler Circuit is a directed graph such that if you start traversing the graph from any node and travel through each edge exactly once you will end up on the starting node.
Note: While traversing a Euler circuit every edge is traversed exactly once. A node can be traversed more than once if needed but an edge cannot be traversed more than once.
Example:
Input:
Output:
1 2
2 5
5 1
2 4
4 3
3 2
Explanation:
The Directed Euler Circuit for the given undirected graph will be:
Approach:

First, we need to make sure the given Undirected Graph is Eulerian or not. If the undirected graph is not Eulerian we cannot convert it to a Directed Eulerian Graph.
 To check it we just need to calculate the degree of every node. If the degree of all nodes is even and not equal to 0 then the graph is Eulerian.

We will be using Depth First Search Traversal to assign the directions.
 While traversing we will set the direction of an edge from parent to child. We will maintain a map to make sure an edge is traversed only once.
Below is the implementation of the above algorithm:
// C++ program to Convert an // Undirected Graph to a // Directed Euler Circuit #include <bits/stdc++.h> using namespace std; vector< int > g[100]; // Array to store degree // of nodes. int deg[100] = { 0 }; // Map to keep a track of // visited edges map<pair< int , int >, int > m1; // Vector to store the edge // pairs vector<pair< int , int > > v; // Function to add Edge void addEdge( int u, int v) { deg[u]++; deg[v]++; g[u].push_back(v); g[v].push_back(u); } // Function to check if graph // is Eulerian or not bool CheckEulerian( int n) { int check = 0; for ( int i = 1; i <= n; i++) { // Checking if odd degree // or zero degree nodes // are present if (deg[i] % 2  deg[i] == 0) { check = 1; break ; } } // If any degree is odd or // any vertex has degree 0 if (check) { return false ; } return true ; } // DFS Function to assign the direction void DirectedEuler( int node, vector< int > g[]) { for ( auto i = g[node].begin(); i != g[node].end(); i++) { // Checking if edge is already // visited if (m1[make_pair(node, *i)]  m1[make_pair(*i, node)]) continue ; m1[make_pair(node, *i)]++; // Storing the edge v.push_back(make_pair(node, *i)); DirectedEuler(*i, g); } } // Function prints the convert // Directed graph void ConvertDirectedEuler( int n, int e) { if (!CheckEulerian(n)) { cout << "NOT POSSIBLE" << endl; return ; } DirectedEuler(1, g); // Printing directed edges for ( auto i = v.begin(); i != v.end(); i++) { cout << (*i).first << " " << (*i).second << endl; } } // Driver code int main() { int N = 5; int E = 6; addEdge(1, 2); addEdge(1, 5); addEdge(5, 2); addEdge(2, 4); addEdge(2, 3); addEdge(4, 3); ConvertDirectedEuler(N, E); } 
1 2 2 5 5 1 2 4 4 3 3 2
Time Complexity: O(( V + E ) * log( E ))
Space Complexity: O(max( V, E ))
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