 x Cycle (graph theory) Encyclopedia
In graph theory
Graph theory
In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...

, the term cycle may refer to a closed path
Path (graph theory)
In graph theory, a path in a graph is a sequence of vertices such that from each of its vertices there is an edge to the next vertex in the sequence. A path may be infinite, but a finite path always has a first vertex, called its start vertex, and a last vertex, called its end vertex. Both of them...

. If repeated vertices
Vertex (graph theory)
In graph theory, a vertex or node is the fundamental unit out of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges , while a directed graph consists of a set of vertices and a set of arcs...

are allowed, it is more often called a closed walk. If the path is a simple path, with no repeated vertices or edges other than the starting and ending vertices, it may also be called a simple cycle, circuit, circle, or polygon; see Cycle graph
Cycle graph
In graph theory, a cycle graph or circular graph is a graph that consists of a single cycle, or in other words, some number of vertices connected in a closed chain. The cycle graph with n vertices is called Cn...

. A cycle in a directed graph is called a directed cycle.

The term cycle may also refer to:
• An element of the binary or integral (or real, complex, etc.) cycle space
Cycle space
In graph theory, an area of mathematics, a cycle space is a vector space defined from an undirected graph; elements of the cycle space represent formal combinations of cycles in the graph....

of a graph. This is the usage closest to that in the rest of mathematics, in particular algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...

. Such a cycle may be called a binary cycle, integral cycle, etc.
• An edge set that has even degree at every vertex; also called an even edge set or, when taken together with its vertices, an even subgraph. This is equivalent to a binary cycle, since a binary cycle is the indicator function of an edge set of this type.

Chordless cycles in a graph are sometimes called graph holes. A graph antihole is the complement
Complement graph
In graph theory, the complement or inverse of a graph G is a graph H on the same vertices such that two vertices of H are adjacent if and only if they are not adjacent in G. That is, to generate the complement of a graph, one fills in all the missing edges required to form a complete graph, and...

of a graph hole.

## Cycle detection

A graph has a cycle if and only if depth-first search
Depth-first search
Depth-first search is an algorithm for traversing or searching a tree, tree structure, or graph. One starts at the root and explores as far as possible along each branch before backtracking....

produces a back edge. This is true for both directed and undirected graphs. In the case of undirected graphs, only O(n) time is required, since at most n-1 edges can be tree edges (where n is the number of vertices). In the case of directed graphs, topological sorting
Topological sorting
In computer science, a topological sort or topological ordering of a directed graph is a linear ordering of its vertices such that, for every edge uv, u comes before v in the ordering...

algorithms will usually detect cycles too, since those are obstacles for topological order to exist. 