Hamiltonian Path/Cycle Problem

Hamiltonian Path in an undirected graph is a path that visits each vertex exactly once. A Hamiltonian cycle is a Hamiltonian Path such that there is an edge (in the graph) from the last vertex to the first vertex of the Hamiltonian Path. Determine whether a given graph contains Hamiltonian Cycle or not. If it contains, then prints the path.

Following are the input and output of the required function.

Input: A 2D array graph[V][V] where V is the number of vertices in graph and graph[V][V] is adjacency matrix representation of the graph. A value graph[i][j] is 1 if there is a direct edge from i to j, otherwise graph[i][j] is 0.

Output: An array path[V] that should contain the Hamiltonian Path. path[i] should represent the ith vertex in the Hamiltonian Path. The code should also return false if there is no Hamiltonian Cycle in the graph. For example, a Hamiltonian Cycle in the following graph is {0, 1, 2, 4, 3, 0}.

(0)--(1)--(2)
 |   / \   |
 |  /   \  | 
 | /     \ |
(3)-------(4)

And the following graph doesn’t contain any Hamiltonian Cycle.

(0)--(1)--(2)
 |   / \   |
 |  /   \  | 
 | /     \ |
(3)      (4) 

Naive Algorithm Generate all possible configurations of vertices and print a configuration that satisfies the given constraints. There will be n! (n factorial) configurations.

while there are untried conflagrations
{
   generate the next configuration
   if ( there are edges between two consecutive vertices of this
      configuration and there is an edge from the last vertex to 
      the first ).
   {
      print this configuration;
      break;
   }
}

Backtracking Algorithm Create an empty path array and add vertex 0 to it.

Add other vertices, starting from the vertex 1. Before adding a vertex, check for whether it is adjacent to the previously added vertex and not already added.

If we find such a vertex, we add the vertex as part of the solution. If we do not find a vertex then we return false.