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There is a simple relation between the two problems. The Hamiltonian path problem for graph G is equivalent to the Hamiltonian cycle problem in a graph '''H''' obtained from G by adding a new vertex and connecting it to all vertices of G. The Hamiltonian cycle problem is a special case of the Traveling Salesman Problem , obtained by setting the distance between two cities to unity if they are adjacent and infinity otherwise. The directed and undirected Hamiltonian cycle problems were two of Karp's 21 NP-complete Problems . Garey and Johnson showed shortly afterwards in 1974 that the directed Hamiltonian cycle problem remains NP-complete for Planar Graph s and the undirected Hamiltonian cycle problem remains NP-complete for Cubic Planar Graph s. QUADRATIC ALGORITHM FOR DIRAC (DENSE) GRAPHS If the degree of every vertex is greater than or equal to v/2, then the problem can be resolved in quadratic time. See {Link without Title} for details. RANDOMIZED ALGORITHM A randomized algorithm for Hamiltonian path that is fast on most graphs is the following: Start from a random vertex, and continue if there is a neighbor not visited. If there are no more unvisited neighbors, and the path formed isn't Hamiltonian, pick a neighbor uniformly at random, and rotate using that neighbor as a pivot. Then, continue the algorithm at the new end of the path. SEE ALSO EXTERNAL LINKS
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