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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 directed cycle graph of length 8. A directed cycle graph is a directed version of a cycle graph, with all the edges being oriented in the same direction. In a directed graph, a set of edges which contains at least one edge (or arc) from each directed cycle is called a feedback arc set.
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 ...
Aperiodic graph, a directed graph in which the cycle lengths have no nontrivial common divisor; Pseudoforest, a directed or undirected graph in which every connected component includes at most one cycle; Cycle graph, a graph that has the structure of a single cycle; Pancyclic graph, a graph that has cycles of all possible lengths; Cycle ...
A directed graph is strongly connected if and only if it has an ear decomposition, a partition of the edges into a sequence of directed paths and cycles such that the first subgraph in the sequence is a cycle, and each subsequent subgraph is either a cycle sharing one vertex with previous subgraphs, or a path sharing its two endpoints with ...
The circuit rank controls the number of ears in an ear decomposition of a graph, a partition of the edges of the graph into paths and cycles that is useful in many graph algorithms. In particular, a graph is 2-vertex-connected if and only if it has an open ear decomposition. This is a sequence of subgraphs, where the first subgraph is a simple ...
An Eulerian cycle, [note 1] also called an Eulerian circuit or Euler tour, in an undirected graph is a cycle that uses each edge exactly once. If such a cycle exists, the graph is called Eulerian or unicursal. [4] The term "Eulerian graph" is also sometimes used in a weaker sense to denote a graph where every vertex has even degree.
In a planar graph, a cycle basis formed by the set of bounded faces of an embedding necessarily has this property: each edge participates only in the basis cycles for the two faces it separates. Conversely, if a cycle basis has at most two cycles per edge, then its cycles can be used as the set of bounded faces of a planar embedding of its graph.