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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 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. Thus, finding a Hamiltonian path cannot be ...
Hamilton's principle states that the true evolution q(t) of a system described by N generalized coordinates q = (q 1, q 2, ..., q N) between two specified states q 1 = q(t 1) and q 2 = q(t 2) at two specified times t 1 and t 2 is a stationary point (a point where the variation is zero) of the action functional [] = ((), ˙ (),) where (, ˙,) is the Lagrangian function for the system.
This argument also gives an algorithm for finding the Hamiltonian path. More efficient algorithms, that require examining only O ( n log n ) {\displaystyle O(n\log n)} of the edges, are known. The Hamiltonian paths are in one-to-one correspondence with the minimal feedback arc sets of the tournament. [ 5 ]
A Hamiltonian cycle in a directed graph is a cycle that passes through each vertex of the graph exactly once. The following Lparse program can be used to find a Hamiltonian cycle in a given directed graph if it exists; we assume that 0 is one of the vertices.
The knight's tour problem is an instance of the more general Hamiltonian path problem in graph theory. The problem of finding a closed knight's tour is similarly an instance of the Hamiltonian cycle problem. Unlike the general Hamiltonian path problem, the knight's tour problem can be solved in linear time. [4]
The requirement of returning to the starting city does not change the computational complexity of the problem; see Hamiltonian path problem. 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.
The Hamiltonian is a function used to solve a problem of optimal control for a dynamical system. It can be understood as an instantaneous increment of the Lagrangian expression of the problem that is to be optimized over a certain time period. [ 1 ]