Consider the following two problems of graph.
1. Given a graph, find if the graph has a cycle that visits every vertex exactly once except the first visited vertex which must be visited again to complete the cycle.
2. Given a graph, find if the graph has a cycle that visits every edge exactly once.
Which of the following is true about above two problems.
a)
Problem 1 belongs NP Complete set and 2 belongs to P
b)
Problem 1 belongs to P set and 2 belongs to NP Complete set
c)
Both problems belong to P set
d)
Both problems belong to NP complete set
Correct answer is option 'A'. Can you explain this answer?

Question

Consider the following two problems of graph.
1. Given a graph, find if the graph has a cycle that visits every vertex exactly once except the first visited vertex which must be visited again to complete the cycle.
2. Given a graph, find if the graph has a cycle that visits every edge exactly once.
Which of the following is true about above two problems.

a)
Problem 1 belongs NP Complete set and 2 belongs to P
b)
Problem 1 belongs to P set and 2 belongs to NP Complete set
c)
Both problems belong to P set
d)
Both problems belong to NP complete set

Correct answer is option 'A'. Can you explain this answer?

Answer

NP Complete Problems in Graph Theory

Problem 1: Hamiltonian Cycle Problem

The Hamiltonian cycle problem is an NP-complete problem in graph theory. The problem is to determine whether a given graph contains a cycle that visits every vertex exactly once, except for the first and last vertices, which are visited twice to form a closed loop.

It is a decision problem that asks whether there exists a Hamiltonian cycle in an undirected graph.

It is NP-complete, which means that it is at least as hard as any other NP problem and is unlikely to have an efficient algorithm for large inputs.

It is a well-known problem in computer science and has been studied extensively.

There are many algorithms for solving this problem, but none of them are known to be efficient for all instances of the problem.

Problem 2: Eulerian Cycle Problem

The Eulerian cycle problem is another NP-complete problem in graph theory. The problem is to determine whether a given graph contains a cycle that visits every edge exactly once.

It is a decision problem that asks whether there exists an Eulerian cycle in a directed or undirected graph.

It is NP-complete, which means that it is at least as hard as any other NP problem and is unlikely to have an efficient algorithm for large inputs.

It is also a well-known problem in computer science and has been studied extensively.

There are many algorithms for solving this problem, but none of them are known to be efficient for all instances of the problem.

Comparison of the two problems

Both problems are related to the existence of cycles in graphs.

The Hamiltonian cycle problem is concerned with visiting every vertex exactly once, while the Eulerian cycle problem is concerned with visiting every edge exactly once.

Both problems are NP-complete, which means that they are among the hardest problems in computer science and are unlikely to have efficient algorithms for large inputs.

Both problems have been studied extensively and have many algorithms for solving them, but none of these algorithms are known to be efficient for all instances of the problems.

Therefore, the correct answer is option A, which states that problem 1 belongs to the NP-complete set, and problem 2 belongs to the P set.

Answer

Problem 1 is Hamiltonian Cycle problem which is a famous NP Complete problem.

Problem 2 is Euler Circuit problem which is solvable in Polynomial time.

Similar Computer Science Engineering (CSE) Doubts

Let S be an NP-complete problem. Q and R are other two problems not known to be NP. Q is polynomial time reducible to S and S is polynomial time reducible to R. Which of the following statements is true?

Consider the following sequence of nodes for the undirected graph given below.a b e f d g ca b e f c g da d g e b c fa d b c g e fA Depth First Search (DFS) is started at node a. The nodes are listed in the order they are first visited. Which all of the above is (ar e) possible output(s)?

For problems X and Y, Y is NP-complete and X reduces to Y in polynomial time. Which of the following is TRUE?

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