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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.
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 ...
For any cycle c in a graph G on m edges, one can form an m-dimensional 0-1 vector that has a 1 in the coordinate positions corresponding to edges in c and a 0 in the remaining coordinate positions. The cycle space C(G) of the graph is the vector space formed by all possible linear
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.
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 cycle rank of a directed acyclic graph is 0, while a complete digraph of order n with a self-loop at each vertex has cycle rank n.Apart from these, the cycle rank of a few other digraphs is known: the undirected path of order n, which possesses a symmetric edge relation and no self-loops, has cycle rank ⌊ ⌋ (McNaughton 1969).
Every cycle basis of a given graph has the same number of cycles, which equals the dimension of its cycle space. This number is called the circuit rank of the graph, and it equals m − n + c {\displaystyle m-n+c} where m {\displaystyle m} is the number of edges in the graph, n {\displaystyle n} is the number of vertices, and c {\displaystyle c ...