# How can the Hamiltonian cycle be reduced?

## How can the Hamiltonian cycle be reduced?

We reduce an instance of Hamiltonian cycle on a graph G = (V,E) to Hamiltonian path in two different ways. (There are many ways to do this.) First reduction: Pick any vertex v ∈ V , split it into two vertices v1 and v2. If (v, u) ∈ E then edges (v1,u) and (v2,u) are included in the new graph.

Is a Hamiltonian cycle a Hamiltonian path?

A Hamiltonian cycle (or Hamiltonian circuit) is a Hamiltonian path that is a cycle. Determining whether such paths and cycles exist in graphs is the Hamiltonian path problem, which is NP-complete.

What is the difference between Hamiltonian cycle and Hamiltonian path?

A Hamiltonian path, also called a Hamilton path, is a graph path between two vertices of a graph that visits each vertex exactly once. If a Hamiltonian path exists whose endpoints are adjacent, then the resulting graph cycle is called a Hamiltonian cycle (or Hamiltonian cycle).

### Is Hamilton path NP hard?

Any Hamiltonian Path can be made into a Hamiltonian Circuit through a polynomial time reduction by simply adding one edge between the first and last point in the path. Therefore we have a reduction, which means that Hamiltonian Paths are in NP Hard, and therefore in NP Complete.

Is Hamiltonian cycle NP-complete problem?

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 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.

#### Why is Hamiltonian cycle NP-hard?

Thus we can say that the graph G’ contains a Hamiltonian Cycle iff graph G contains a Hamiltonian Path. 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.

What is Hamiltonian cycle give example?

How to reduce Hamiltonian cycle to Hamiltonian path?

Given a graph G of which we need to find Hamiltonian Cycle, for a single edge e = { u, v } add new vertices u ′ and v ′ such that u ′ is connected only to u and v ′ is connected only to v to give a new graph G e. G e has a Hamiltonian path if and only if G has a Hamiltonian cycle with the edge e = { u, v }.

## What kind of problem is the Hamiltonian path problem?

Hamiltonian path problem. In the mathematical field of graph theory the Hamiltonian path problem and the Hamiltonian cycle problem are problems of determining whether a Hamiltonian path (a path in an undirected or directed graph that visits each vertex exactly once) or a Hamiltonian cycle exists in a given graph (whether directed or undirected).

When does a graph have no Hamiltonian cycle?

If all graphs have no Hamiltonian path, then G has no Hamiltonian cycle. If at least one G e has a Hamiltonian path, then G has a Hamiltonian cycle which contains the edge e. I think you wrong about the reduction from HC to HP (answer #1).

Is the directed and undirected Hamiltonian cycle problem NP complete?

The directed and undirected Hamiltonian cycle problems were two of Karp’s 21 NP-complete problems. They remain NP-complete even for special kinds of graphs, such as: cubic subgraphs of the square grid graph. However, for some special classes of graphs, the problem can be solved in polynomial time:

16/03/2020