## Is directed Hamiltonian cycle NP-complete?

Therefore, any instance of the Hamiltonian Cycle problem can be reduced to an instance of the Hamiltonian Path problem. Thus, the Hamiltonian Cycle is NP-Hard. Conclusion: Since, the Hamiltonian Cycle is both, a NP-Problem and NP-Hard. Therefore, it is a NP-Complete problem.

### What is a directed Hamiltonian cycle?

Definition 1 A Hamiltonian path P in a directed graph G(V,H) is a simple path which contains every vertex of the graph exactly once. A Hamiltonian cycle C in a graph G(V,E) is a simple cycle which contains every vertex exactly once. A graph is called Hamiltonian if it has a Hamiltonian cycle.

#### How do you prove that Hamiltonian cycle NP problem?

To prove this, one way is to show that the Hamiltonian cycle ≤p TSP (as we know that the Hamiltonian cycle problem is NP Complete). Assume G = (V, E) to be an instance of the Hamiltonian cycle. Now, assume that a Hamiltonian cycle H exists in G. The cost of each edge in H is 0 in G’ as each edge belongs to E.

How many Hamiltonian cycles are in a complete graph?

Theorem. For all n≥3, the number of distinct Hamilton cycles in the complete graph Kn is (n−1)! 2.

What is the main difference between Hamiltonian cycle problem and Travelling salesman problem?

Answer. One difference is that the traveling salesman problem is a Hamiltonian cycle. Another difference is that the traveling salesman problem is used to find a path that contains permutationof every node in a graph, and it is a NP-complete problem and the shortest path is known polynomial-time.

## Do all complete graphs have Hamiltonian cycles?

Every complete graph with more than two vertices is a Hamiltonian graph. This follows from the definition of a complete graph: an undirected, simple graph such that every pair of nodes is connected by a unique edge. The graph of every platonic solid is a Hamiltonian graph.

### How many cycles are in a complete graph?

Actually a complete graph has exactly (n+1)! cycles which is O(nn).

#### Is complete graph Hamiltonian?

Is Hamiltonian cycle and Travelling salesman NP-complete?

The verification algorithm is possible if a tour can be guessed which is of length of k. This verification can be done in polynomial time. Therefore, Hamiltonian cycle is a NP-Complete problem. Assign k =n.

Is Hamiltonian cycle A & travelling salesman problem NP-complete?

The edges it uses are from E (given the cost function we created), hence we can say for sure that there is a Hamiltonian Cycle in G. Given that Hamiltonian Cycle is NP-Complete, the decision version of TSP is NP-Complete, hence TSP is NP-Hard.

## Is Sudoku a NP?

Sudoku is NP-complete when generalized to a n × n grid however a standard 9 × 9 Sudoku is not NP- complete.

### How to prove the Hamiltonian cycle is NP-hard?

This can be done in polynomial time, that is O (V +E) using the following strategy for graph G (V, E): In order to prove the Hamiltonian Cycle is NP-Hard, we will have to reduce a known NP-Hard problem to this problem. We will carry out a reduction from the Hamiltonian Path problem to the Hamiltonian Cycle problem.

#### What is Hamiltonian cycle in independent set?

Hamiltonian Cycle: A cycle in an undirected graph G = (V, E) which traverses every vertex exactly once. An instance of the problem is an input specified to the problem. An instance of the Independent Set problem is a graph G (V, E), and the problem is to check whether the graph can have a Hamiltonian Cycle in G.

Is the Hamiltonian path of a directed graph NP-complete?

Hamiltonian Path or HAMPATH in a directed graph G is a directed path that goes through each node exactly once. We Consider the problem of testing whether a directed graph contain a Hamiltonian path connecting two specified nodes, i.e. To prove HAMPATH is NP-Complete we have to prove that HAMPATH is in NP.

How to prove hampath is NP-complete?

To prove HAMPATH is NP-Complete we have to prove that HAMPATH is in NP. To prove HAMPATH is in NP we must have a polynomial-time verifier.