## 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.

Is the Hamiltonian cycle problem NP-complete?

Conclusion: Since, the Hamiltonian Cycle is both, a NP-Problem and NP-Hard. Therefore, it is a NP-Complete problem.

How do you prove the Hamiltonian cycle?

Theorem: Let G be a simple graph with at least 3 vertices. If every vertex of G has degree ≥ |V (G)|/2, then G has a Hamiltonian cycle.

### What is Hamiltonian cycle with example?

A dodecahedron ( a regular solid figure with twelve equal pentagonal faces) has a Hamiltonian cycle. A Hamiltonian cycle is a closed loop on a graph where every node (vertex) is visited exactly once.

How do you prove TSP is NP-complete?

To prove TSP is NP-Complete, first we have to prove that TSP belongs to NP. In TSP, we find a tour and check that the tour contains each vertex once. Then the total cost of the edges of the tour is calculated. Finally, we check if the cost is minimum.

What is a Hamiltonian path and Hamiltonian circuit explain with the example?

A Hamilton Path is a path that goes through every Vertex of a graph exactly once. A Hamilton Circuit is a Hamilton Path that begins and ends at the same vertex. Hamilton Path Hamilton Circuit. *notice that not all edges need to be used.

## What are NP-complete problems give examples?

NP-complete problem, any of a class of computational problems for which no efficient solution algorithm has been found. Many significant computer-science problems belong to this class—e.g., the traveling salesman problem, satisfiability problems, and graph-covering problems.

How do you prove that Hamiltonian cycle is NP complete?

PROOFS OF NP-COMPLETENESS 689 (2) Hamiltonian Cycle (for Directed Graphs) To prove that Hamiltonian Cycle (for Directed Graphs) is NP-complete, we will reduce Exact Cover to it: Exact Cover ≤

How to reduce Hamiltonian path problem to Hamiltonian cycle problem?

We will carry out a reduction from the Hamiltonian Path problem to the Hamiltonian Cycle problem. Every instance of the Hamiltonian Path problem consisting of a graph G = (V, E) as the input can be converted to Hamiltonian Cycle problem consisting of graph G’ = (V’, E’).

### Is the Hamiltonian cycle NP-hard or NP-soft?

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.

How to validate the Certificate of Hamiltonian cycle?

The certificate is a sequence of vertices forming Hamiltonian Cycle in the graph. We can validate this solution by verifying that all the vertices belong to the graph and each pair of vertices belonging to the solution are adjacent. This can be done in polynomial time, that is O (V +E) using the following strategy for graph G (V, E):