HAMILTONIAN CYCLE USING BACKTRACKING

Hamiltonian Path Backtracking:

To solve the Hamiltonian path problem using backtracking, we can follow these steps:

1. Start with an empty path and an empty set of visited vertices.
2. Choose a starting vertex and add it to the path and the visited set.
3. For each neighbor of the last vertex in the path that has not been visited, add the neighbor to the path and the visited set.
4. If the path contains all vertices in the graph, return the path as a solution.
5. If there are no unvisited neighbors for the last vertex in the path, backtrack by removing the last vertex from the path and the visited set.
6. Repeat steps 3-5 until all solutions have been found or no solution exists.

The algorithm explores all possible paths in the graph and backtracks when it reaches a dead end. The worst-case time complexity of this algorithm is O(n!), where n is the number of vertices in the graph. However, the actual running time will depend on the structure of the graph and the choice of starting vertex.