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Cycle detection is the problem of finding i and j, given f and x 0. Several algorithms are known for finding cycles quickly and with little memory. Robert W. Floyd 's tortoise and hare algorithm moves two pointers at different speeds through the sequence of values until they both point to equal values.
The problem of finding a Hamiltonian cycle or path is in FNP; the analogous decision problem is to test whether a Hamiltonian cycle or path exists. The directed and undirected Hamiltonian cycle problems were two of Karp's 21 NP-complete problems. They remain NP-complete even for special kinds of graphs, such as: bipartite graphs, [12]
The directed feedback vertex set problem can still be solved in time O*(1.9977 n), where n is the number of vertices in the given directed graph. [5] The parameterized versions of the directed and undirected problems are both fixed-parameter tractable .
A special case of this method is the use of the modular product of graphs to reduce the problem of finding the maximum common induced subgraph of two graphs to the problem of finding a maximum clique in their product. [7] In automatic test pattern generation, finding cliques can help to bound the size of a test set. [8]
Another related problem is the bottleneck travelling salesman problem: Find a Hamiltonian cycle in a weighted graph with the minimal weight of the weightiest edge. A real-world example is avoiding narrow streets with big buses. [15] The problem is of considerable practical importance, apart from evident transportation and logistics areas.
The figure below presents an example of the execution of the algorithm. In iteration =, all the three vertices detect the cycle (,,). The algorithm ensures that the cycle is reported only once by emitting the detected cycle only from the vertex with the least identifier value in the ordered sequence, which is the vertex 2 in the example.
In graph theory, a cycle in a graph is a non-empty trail in which only the first and last vertices are equal. A directed cycle in a directed graph is a non-empty directed trail in which only the first and last vertices are equal. A graph without cycles is called an acyclic graph. A directed graph without directed cycles is called a directed ...
A Hamiltonian cycle, Hamiltonian circuit, vertex tour or graph cycle is a cycle that visits each vertex exactly once. A graph that contains a Hamiltonian cycle is called a Hamiltonian graph . Similar notions may be defined for directed graphs , where each edge (arc) of a path or cycle can only be traced in a single direction (i.e., the vertices ...