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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 significantly slower (in the worst case, as a function of the number of vertices) than finding a ...
A graph that contains a Hamiltonian path is called a traceable graph. A graph is Hamiltonian-connected if for every pair of vertices there is a Hamiltonian path between the two vertices. A Hamiltonian cycle, Hamiltonian circuit, vertex tour or graph cycle is a cycle that visits each vertex exactly once.
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 ]
Dirac's theorem on Hamiltonian cycles, the statement that an n-vertex graph in which each vertex has degree at least n/2 must have a Hamiltonian cycle; Dirac's theorem on chordal graphs, the characterization of chordal graphs as graphs in which all minimal separators are cliques
For, any unweighted graph can be transformed into a metric space by setting its edge weights to 1 and setting the distance between all nonadjacent pairs of vertices to 2. An approximation with ratio better than 2 in this metric space could be used to determine whether the original graph contains a Hamiltonian cycle, an NP-complete problem. [6]
Ore's theorem is a generalization of Dirac's theorem that, when each vertex has degree at least n/2, the graph is Hamiltonian. For, if a graph meets Dirac's condition, then clearly each pair of vertices has degrees adding to at least n. In turn Ore's theorem is generalized by the Bondy–Chvátal theorem.
William Rowan Hamilton, the inventor of the icosian game. At the time of his invention of the icosian game, William Rowan Hamilton was the Andrews Professor of Astronomy at Trinity College Dublin and Royal Astronomer of Ireland, and was already famous for his work on Hamiltonian mechanics and his invention of quaternions. [9]
The Hamiltonian completion problem is to find the minimal number of edges to add to a graph to make it Hamiltonian. 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 ).