This assignment continues to explore NP-Completeness and NP-Complete problems.

Homework Problems

1. Undirected Hamiltonian Paths (12 pts)

2. Hamiltonian Cycles (12 pts)

3. Making Hamiltonian Paths (11 pts)

4. README (1 point)

Total: 36 points

Submitting

Submit your solution to this assignment in Gradescope hw12. Please assign each page to the correct problem and make sure your solutions are legible.

A submission must also include a README containing the required information.

1 Undirected Hamiltonian Paths

Prove that UHAMPATH (from lecture) is NP-Complete. Start with the ideas from class. Make sure to include all the required parts of the proof as described in lecture.

2 Hamiltonian Cycles

Recall that a cycle in a graph (see Sipser Ch 0) is a path that starts and ends at the same vertex. Also, a Hamiltonian path is a path that touches every vertex in the graph.

Prove that the following language is NP-Complete.

HCYCLE={G|G is a directed graph with a Hamiltonian cycle}

Make sure to include all the required parts of the proof.

3 Making Hamiltonian Paths

Recall that a Hamiltonian path is a path that touches every vertex in the graph.

Prove that the following language is NP-Complete.

HMAKE={?G,k?|G is a directed graph that has a Hamiltonian path if k edges are added to it}

Make sure to include all the required parts of the proof.

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