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In computer science, cycle detection or cycle finding is the algorithmic problem of finding a cycle in a sequence of iterated function values. For any function f that maps a finite set S to itself, and any initial value x 0 in S , the sequence of iterated function values
Therefore, the special case of the zero-weight cycle problem, on graphs with no negative cycle, has a polynomial-time algorithm. [1] In contrast, for graphs that contain negative cycles, detecting a simple cycle of weight exactly 0 is an NP-complete problem. [1] This is true even when the weights are integers of polynomial magnitude.
The Rocha–Thatte algorithm is a general algorithm for detecting cycles in a directed graph by message passing among its vertices, based on the bulk synchronous message passing abstraction. This is a vertex-centric approach in which the vertices of the graph work together for detecting cycles.
The problems of finding a Hamiltonian path and a Hamiltonian cycle can be related as follows: In one direction, the Hamiltonian path problem for graph G can be related to the Hamiltonian cycle problem in a graph H obtained from G by adding a new universal vertex x, connecting x to all vertices of G.
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]
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 ...
The problem is clearly NP-hard in the general case (since its solution gives an answer to the NP-complete problem of determining whether a given graph has a Hamiltonian cycle). The associated decision problem of determining whether K edges can be added to a given graph to produce a Hamiltonian graph is NP-complete.