Check if a Directed Graph Contains an Eulerian Cycle in C++

The Euler cycle/circuit is a path; by which we can visit every edge exactly once. We can use the same vertices for multiple times. The Euler Circuit is a special type of Euler path. When the starting vertex of the Euler path is also connected with the ending vertex of that path, then it is called the Euler Circuit.

To check whether a graph is Eulerian or not, we have to check two conditions −

• The graph must be connected.

• The in-degree and out-degree of each vertex must be same.

Input − Adjacency matrix of the graph.

Output − Euler Circuit Found

Algorithm

traverse(u, visited)

Input − The start node u and the visited node to mark which node is visited.

Output − Traverse all connected vertices.

Begin
   mark u as visited
   for all vertex v, if it is adjacent with u, do
      if v is not visited, then
         traverse(v, visited)
   done
End

isConnected(graph)

Input − The graph.

Output − True if the graph is connected.

Begin
   define visited array
   for all vertices u in the graph, do
      make all nodes unvisited
      traverse(u, visited)
      if any unvisited node is still remaining, then
         return false
   done
   return true
End

isEulerCircuit(Graph)

Input − The given Graph.

Output − True when one Euler circuit is found.

Begin
   if isConnected() is false, then
      return false
   define list for inward and outward edge count for each node
   for all vertex i in the graph, do
      sum := 0
      for all vertex j which are connected with i, do
         inward edges for vertex i increased
         increase sum
      done
      number of outward of vertex i is sum
   done
   if inward list and outward list are same, then
       return true
    otherwise return false
End

Example Code

#include<iostream>
#include<vector>
#define NODE 5
using namespace std;
int graph[NODE][NODE] = {{0, 1, 0, 0, 0},
   {0, 0, 1, 0, 0},
   {0, 0, 0, 1, 1},
   {1, 0, 0, 0, 0},
   {0, 0, 1, 0, 0}};
void traverse(int u, bool visited[]) {
   visited[u] = true;     //mark v as visited
   for(int v = 0; v<NODE; v++) {
      if(graph[u][v]) {
         if(!visited[v])
            traverse(v, visited);
      }
   }
}
bool isConnected() {
   bool *vis = new bool[NODE];
   //for all vertex u as start point, check whether all nodes are visible or not
   for(int u; u < NODE; u++) {
      for(int i = 0; i<NODE; i++)
         vis[i] = false;     //initialize as no node is visited
         traverse(u, vis);
      for(int i = 0; i<NODE; i++) {
         if(!vis[i])     //if there is a node, not visited by traversal, graph is not connected
            return false;
      }
   }
   return true;
}
bool isEulerCircuit() {
   if(isConnected() == false) {     //when graph is not connected
      return false;
   }
   vector<int> inward(NODE, 0), outward(NODE, 0);
   for(int i = 0; i<NODE; i++) {
      int sum = 0;
      for(int j = 0; j<NODE; j++) {
         if(graph[i][j]) {
            inward[j]++;     //increase inward edge for destination
            vertex
            sum++;    //how many outward edge
         }
      }
      outward[i] = sum;
   }
   if(inward == outward)      //when number inward edges and outward edges
      for each node is same
         return true;
   return false;
}
int main() {
   if(isEulerCircuit())
      cout << "Euler Circuit Found.";
   else
     cout << "There is no Euler Circuit.";
}

Output

Euler Circuit Found.