Graph Theory Problems

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Study Notes

Graph Problems

• 3-COLOR problem: Given a planar map, can it be colored using 3 colors so that no adjacent regions have the same color?
• VERTEX COVER problem: Given a graph G(V,E) and a positive integer k, find a vertex cover of size at most k
• HAMILTONIAN CYCLE problem: Given an undirected graph, is there a Hamiltonian cycle in this graph?

NP-Completeness

• NPC problem: A problem B is NPC if B is in NP and every problem in NP can be reduced to B in polynomial time
• To prove that a problem B is NPC: B is in NP, and there is a polynomial transformation from a known NPC problem A to B

P and NP

• P: a set of decision problems that can be solved in polynomial time
• NP: a set of decision problems that can be verified in polynomial time
• NP is a subset of NP and as hard as other problems in NP
• NP-Hard: the set of problems that every NP problem can be reduced to one of it

Decision Problems

• Decision Problem: a problem of determining an answer to a class of yes/no questions
• Examples: given a graph G, vertices u and v, an integer k, does a path exist from u to v consisting of at most k edges?; does a graph have a path that goes through every node exactly once?; is the number x prime?
• A solution to a decision problem is given by an algorithm that answers yes or no

Optimization Problems

• Optimization Problem: the answer of the problem is a feasible solution with the best (minimum or maximum) value
• Examples: knapsack problem, single-source shortest path, maximum flow, dynamic programming, divide-and-conquer, prune-and-search

Test your knowledge of graph theory concepts, including graph coloring and vertex cover problems. Solve exercises and quizzes to improve your understanding of these fundamental computer science topics.

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