# C++ Program to Check Whether a Hamiltonian Cycle or Path Exists in a Given Graph

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A Hamiltonian cycle is a Hamiltonian Path such that there is an edge (in graph) from the last vertex to the first vertex of the Hamiltonian Path. It is in an undirected graph is a path that visits each vertex of the graph exactly once.

## Functions and purposes:

Begin
1.function isSafe() is used to check for whether it is adjacent to the previously added vertex and already not added.
2. function hamiltonianCycle() solves the hamiltonian problem.
3. function hamCycle() uses hamiltonianCycle() to solve the hamiltonian problem.
It returns false if there is no Hamiltonian Cycle possible, otherwise return true and prints the path.
End

## Example

#include <iostream>
#include <cstdio>
#include <cstdlib>
#define N 5
using namespace std;
void displaytheSolution(int path[]);
bool isSafe(int n, bool g[N][N], int path[], int pos) {
if (g [path[pos-1]][n] == 0)
return false;
for (int i = 0; i < pos; i++)
if (path[i] == n)
return false;
return true;
}
bool hamiltonianCycle(bool g[N][N], int path[], int pos) {
//If all vertices are included in Hamiltonian Cycle
if (pos == N) {
if (g[ path[pos-1] ][ path ] == 1)
return true;
else
return false;
}
for (int n = 1; n < N; n++) {
if (isSafe(n, g, path, pos)) //Check if this vertex can be added to Hamiltonian Cycle
{
path[pos] = n;
//recur to construct rest of the path
if (hamiltonianCycle (g, path, pos+1) == true)
return true;
path[pos] = -1; //remove vertex if it doesn’t lead to the solution
}
}
return false;
}
bool hamCycle(bool g[N][N]) {
int *path = new int[N];
for (int i = 0; i < N; i++)
path[i] = -1;
//put vertex 0 as the first vertex in the path.
If there is a Hamiltonian Cycle, then the path can be started from any point
//of the cycle as the graph is undirected
path = 0;
if (hamiltonianCycle(g, path, 1) == false) {
cout<<"\nCycle does not exist"<<endl;
return false;
}
displaytheSolution(path);
return true;
}
void displaytheSolution(int p[]) {
cout<<"Cycle Exists:";
cout<<" Following is one Hamiltonian Cycle \n"<<endl;
for (int i = 0; i < N; i++)
cout<<p[i]<<" ";
cout<< p<<endl;
}
int main() {
bool g[N][N] = {
{0, 1, 0, 1, 1},
{0, 0, 1, 1, 0},
{0, 1, 0, 1, 1},
{1, 1, 1, 0, 1},
{0, 1, 1, 0, 0},
};
hamCycle(g);
return 0;
}

## Output

Cycle Exists: Following is one Hamiltonian Cycle
0 4 1 2 3 0