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Since there is no way of partitioning the faces into two subsets that produce a sum obeying Grinberg's theorem, there can be no Hamiltonian cycle. [1] For instance, for the graph in the figure, all the bounded faces have 5 or 8 sides, but the unbounded face has 9 sides, so it satisfies this condition on numbers of sides and is not Hamiltonian.
The Hamiltonian in this case is known as a sub-Riemannian Hamiltonian. Every such Hamiltonian uniquely determines the cometric, and vice versa. This implies that every sub-Riemannian manifold is uniquely determined by its sub-Riemannian Hamiltonian, and that the converse is true: every sub-Riemannian manifold has a unique sub-Riemannian Hamiltonian
A fundamental cycle basis may be formed from any spanning tree or spanning forest of the given graph, by selecting the cycles formed by the combination of a path in the tree and a single edge outside the tree. Alternatively, if the edges of the graph have positive weights, the minimum weight cycle basis may be constructed in polynomial time.
This is a list of recurring cycles. See also Index of wave articles ... sequence – Fourier series – Frequency domain – Frequency spectrum – Hamiltonian ...
The line graphs of cubic graphs are 4-regular and have a Hamiltonian decomposition if and only if the underlying cubic graph has a Hamiltonian cycle. [12] [13] As a consequence, Hamiltonian decomposition remains NP-complete for classes of graphs that include line graphs of hard instances of the Hamiltonian cycle problem. For instance ...
It implies that computing, up to the -th power of , the Hamiltonian cycle polynomial of a unitary n×n-matrix over the infinite extension of any ring of characteristic q (not necessarily prime) by the formal variable is a # P-complete problem if isn't 2 and computing the Hamiltonian cycle polynomial of a -semi-unitary matrix (i.e. an n×n ...
In planarity testing, H is a cycle and a peripheral cycle is a cycle with at most one bridge; it must be a face boundary in any planar embedding of its graph. 3. A bridge of a cycle can also mean a path that connects two vertices of a cycle but is shorter than either of the paths in the cycle connecting the same two vertices.
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 ...