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.

Algorithm

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

dfs(prev, start, visited)
Input: The pervious and start vertex to perform DFS, and visited list.
Output: Count the number of nodes after DFS.
Begin
   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
   done
   return count
End

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

fleuryAlgorithm(start)
Input: The starting vertex.
Output: Display the Euler path or circuit.
Begin
   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 
Example
#include
#include
#include
#define NODE 8

using namespace std;
int graph[NODE][NODE] = {
   {0,1,1,0,0,0,0,0},
   {1,0,1,1,1,0,0,0},
   {1,1,0,1,0,1,0,0},
   {0,1,1,0,0,0,0,0},
   {0,1,0,0,0,1,1,1},
   {0,0,1,0,1,0,1,1},
   {0,0,0,0,1,1,0,0},
   {0,0,0,0,1,1,0,0}
};
int tempGraph[NODE][NODE];
int findStartVert() {
   for(int i = 0; i1) {
      return false; //the edge is not forming bridge
   }
   return true; //edge forming a bridge
}
int edgeCount() {
   int count = 0;
   for(int i = 0; i
Output
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