Fleury’s Algorithm

Fleury’s Algorithm is used to display the Euler path or Euler circuit from a given graph. In this algorithm, starting from one edge, it tries to move other adjacent vertices by removing the previous vertices. Using this trick, the graph becomes simpler in each step to find the Euler path or circuit.

We have to check some rules to get the path or circuit −

  • The graph must be a Euler Graph.

  • When there are two edges, one is bridge, another one is non-bridge, we have to choose non-bridge at first.

Choosing of starting vertex is also tricky, we cannot use any vertex as starting vertex, if the graph has no odd degree vertices, we can choose any vertex as start point, otherwise when one vertex has odd degree, we have to choose that one first.


Input: The given graph.
Output: Find the starting vertex to start algorithm.
   for all vertex i, in the graph, do
      deg := 0
      for all vertex j, which are adjacent with i, do
         deg := deg + 1
      if deg is odd, then
         return i
   when all degree is even return 0

dfs(prev, start, visited)
Input: The pervious and start vertex to perform DFS, and visited list.
Output: Count the number of nodes after DFS.
   count := 1
   visited[start] := true
   for all vertex b, in the graph, do
      if prev is not u, then
         if u is not visited, then
            if start and u are connected, then
               count := count + dfs(start, u, visited)
            end if
         end if
      end if
   return count

isBridge(u, v)
Input: The start and end node.
Output: True when u and v are forming a bridge.
   deg := 0
   for all vertex i which are adjacent with v, do
      deg := deg + 1
   if deg > 1, then
      return false
   return true

Input: The starting vertex.
Output: Display the Euler path or circuit.
   edge := get the number of edges in the graph
   //it will not initialize in next
   recursion call
   v_count = number of nodes
   //this will not initialize in next recursion call
   for all vertex v, which are adjacent with start, do
      make visited array and will with false value
      if isBridge(start, v), then decrease v_count by 1
      cnt = dfs(start, v, visited)
      if difference between cnt and v_count <= 2, then
         print the edge (start →‡ v)
         if isBridge(v, start), then decrease v_count by 1
         remove edge from start and v
         decrease edge by 1
      end if


#define NODE 8

using namespace std;
int graph[NODE][NODE] = {
int tempGraph[NODE][NODE];
int findStartVert() {
   for(int i = 0; i<NODE; i++) {
      int deg = 0;
      for(int j = 0; j<NODE; j++) {
            deg++; //increase degree, when connected edge found
      if(deg % 2 != 0) //when degree of vertices are odd
      return i; //i is node with odd degree
   return 0; //when all vertices have even degree, start from 0
int dfs(int prev, int start, bool visited[]){
   int count = 1;
   visited[start] = true;
   for(int u = 0; u<NODE; u++){
      if(prev != u){
               count += dfs(start, u, visited);
   return count;
bool isBridge(int u, int v) {
   int deg = 0;
   for(int i = 0; i<NODE; i++)
   if(deg>1) {
      return false; //the edge is not forming bridge
   return true; //edge forming a bridge
int edgeCount() {
   int count = 0;
   for(int i = 0; i<NODE; i++)
      for(int j = i; j<NODE; j++)
   return count;
void fleuryAlgorithm(int start) {
   static int edge = edgeCount();
   static int v_count = NODE;
   for(int v = 0; v<NODE; v++) {
      if(tempGraph[start][v]) {
         bool visited[NODE] = {false};
         if(isBridge(start, v)){
         int cnt = dfs(start, v, visited);
         if(abs(v_count-cnt) <= 2){
            cout << start << "--" << v << " ";
            if(isBridge(v, start)){
            tempGraph[start][v] = tempGraph[v][start] = 0; //remove edge from graph
int main() {
   for(int i = 0; i<NODE; i++) //copy main graph to tempGraph
   for(int j = 0; j<NODE; j++)
      tempGraph[i][j] = graph[i][j];
   cout << "Euler Path Or Circuit: ";


Euler Path Or Circuit: 0--1 1--2 2--3 3--1 1--4 4--5 5--6 6--4 4--7 7--5 5--2 2--0

Updated on: 16-Jun-2020

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