# Fleury’s Algorithm

Data StructureAlgorithmsGraph Algorithms

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 <= 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
fleuryAlgorithm(v)
end if
done
End

## Example

#include<iostream>
#include<vector>
#include<cmath>
#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; i<NODE; i++) {
int deg = 0;
for(int j = 0; j<NODE; j++) {
if(tempGraph[i][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){
if(!visited[u]){
if(tempGraph[start][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(tempGraph[v][i])
deg++;
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++)
if(tempGraph[i][j])
count++;
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)){
v_count--;
}
int cnt = dfs(start, v, visited);
if(abs(v_count-cnt) <= 2){
cout << start << "--" << v << " ";
if(isBridge(v, start)){
v_count--;
}
tempGraph[start][v] = tempGraph[v][start] = 0; //remove edge from graph
edge--;
fleuryAlgorithm(v);
}
}
}
}
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: ";
fleuryAlgorithm(findStartVert());
}

## 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