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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?
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Consider the following two problems of graph.1. Given a graph, find if...
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.
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Consider the following two problems of graph.1. Given a graph, find if...
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.
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Understanding the Complexity ClassesIn computational theory, problems are categorized into complexity classes based on the resources required to solve them. Here, we focus on four specific problems: PATH, HAMPATH, SAT, and 3SAT, to determine which belong to class P.What is Class P?- Class P consists of decision problems that can be solved by a deterministic Turing machine in polynomial time.- If a problem is in P, it means there exists an algorithm that can solve it efficiently for all input sizes.Analysis of Each Problem- PATH: - The problem asks whether there exists a path between two vertices in a graph. - This can be solved using Depth-First Search (DFS) or Breadth-First Search (BFS), both of which run in polynomial time. - Conclusion: PATH is in P.- HAMPATH: - This problem involves determining whether there is a Hamiltonian path in a graph (a path that visits each vertex exactly onc e). - HAMPATH is NP-complete, meaning it is not known to be solvable in polynomial time. - Conclusion: HAMPATH is not in P.- SAT (Satisfiability): - SAT asks whether a boolean formula can be satisfied by some assignment of truth values. - While SAT is NP-complete, it was proven to be in NP, and due to advancements, it can be solved in polynomial time for specific cases. - Conclusion: SAT is not in P generally.- 3SAT: - This is a specific case of SAT where the formula is in conjunctive normal form with exactly three literals per clause. - 3SAT is also NP-complete and does not have known polynomial-time solutions. - Conclusion: 3SAT is not in P.Final ConclusionBased on the analyses, the problems that belong to class P are:- Option A: SAT - Not generally in P.- Option C: PATH - In P.Thus, only PATH is confirmed to be in P.

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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 Pb)Problem 1 belongs to P set and 2 belongs to NP Complete setc)Both problems belong to P setd)Both problems belong to NP complete setCorrect answer is option 'A'. Can you explain this answer?
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