Eulerian Path and Circuit

The Euler path 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 detect the path and circuit, we have to follow these conditions −

• The graph must be connected.
• When exactly two vertices have odd degree, it is a Euler Path.
• Now when no vertices of an undirected graph have odd degree, then it is a Euler Circuit.

Input and Output

Input:
Adjacency matrix of a graph.
0 1 1 1 0
1 0 1 0 0
1 1 0 0 0
1 0 0 0 1
0 0 0 1 0

Output:
The graph has an Eulerian path.

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

isEulerian(Graph)

Input − The given Graph.

Output − Returns 0, when not Eulerian, 1 when it has a Euler path, 2 when Euler circuit found

Begin
   if isConnected() is false, then
      return false
   define list of degree for each node
   oddDegree := 0

   for all vertex i in the graph, do
      for all vertex j which are connected with i, do
         increase degree
      done
      if degree of vertex i is odd, then
         increase dooDegree
   done

   if oddDegree > 2, then
      return 0
   if oddDegree = 0, then
      return 2
   else
      return 1
End

Example

#include<iostream>
#include<vector>
#define NODE 5
using namespace std;

int graph[NODE][NODE] = {
   {0, 1, 1, 1, 0},
   {1, 0, 1, 0, 0},
   {1, 1, 0, 0, 0},
   {1, 0, 0, 0, 1},
   {0, 0, 0, 1, 0}
};
                               
/* int graph[NODE][NODE] = {
   {0, 1, 1, 1, 1},
   {1, 0, 1, 0, 0},
   {1, 1, 0, 0, 0},
   {1, 0, 0, 0, 1},
   {1, 0, 0, 1, 0}
};
*/    //uncomment to check Euler Circuit
                               
/* int graph[NODE][NODE] = {
   {0, 1, 1, 1, 0},
   {1, 0, 1, 1, 0},
   {1, 1, 0, 0, 0},
   {1, 1, 0, 0, 1},
   {0, 0, 0, 1, 0}
};
*/    //Uncomment to check Non Eulerian Graph
               
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;
}

int isEulerian() {
   if(isConnected() == false)    //when graph is not connected
      return 0;
   vector<int> degree(NODE, 0);
   int oddDegree = 0;

   for(int i = 0; i<NODE; i++) {
      for(int j = 0; j<NODE; j++) {
         if(graph[i][j])
            degree[i]++;    //increase degree, when connected edge found
      }

      if(degree[i] % 2 != 0)    //when degree of vertices are odd
         oddDegree++; //count odd degree vertices
   }

   if(oddDegree > 2)    //when vertices with odd degree greater than 2
      return 0;
         
   return (oddDegree)?1:2;    //when oddDegree is 0, it is Euler circuit, and when 2, it is Euler path
}

int main() {
   int check;
   check = isEulerian();

   switch(check) {
      case 0: cout << "The graph is not an Eulerian graph.";
         break;
      case 1: cout << "The graph has an Eulerian path.";
         break;
      case 2: cout << "The graph has a Eulerian circuit.";
         break;
   }
}

Output

The graph has an Eulerian path.