Cycle (graph theory)

A cycle or circuit is a path in a graph where the start and end nodes are equal in graph theory. A cyclic graph is a graph with at least one cycle. Algorithms for finding cycles in a graph is a modified topological sort or a modified depth-first search.

  • 2.1 Cyclic Graph
  • 2.2 Panzyklischer graph

Definitions

Cycle

Is a graph, it is a way to cycle basis when

Applies. In a cycle, so must start and end nodes of the path match. A cycle in a directed graph is called directed and undirected cycle in an undirected graph cycle.

Circle

Corresponding thereto is called a cycle graph in a circuit when a path is. A circle is thus a cycle in which only the start and end nodes are the same, it is therefore additionally

For with. A circle in a directed graph is called directed and undirected circuit in an undirected graph circle. An edge connecting two nodes of a circuit, but is not itself part of the circle is the chord of the circle.

Length

In graphs without edge weights is the length of a cycle or circle. So clearly you count the number of associated edges. In an edge-weighted graph, the length of a cycle or loop is the sum of weights of all edges associated edge.

Special graphs

Cyclic graph

A graph with at least one cycle is called cyclic. Graphs without cycles are called acyclic or forest. A cycle or circuit is called trivial if it contains less than three nodes. Trivial circuits or cycles are generally not considered in the analysis of the graph. A circle containing exactly three nodes triangle is called. A graph without triangle is then called triangle -free. As a girth of a graph is defined as the length of a shortest non-trivial circle. If the graph has no cycle, as is given to the waist to infinity. The simplest cyclic graphs are the circle graph.

Panzyklischer graph

Is a graph kantenpanzyklisch if any edge is on a circle of length for all. Is a graph knotenpanzyklisch, if each node is on a circle of length for all. A graph is called pancyclic if it has a cycle of length for all. Kantenpanzyklische graphs are thus knotenpanzyklisch and knotenpanzyklische graph also pancyclic. Panzyklische graphs are particular hamiltonian.

Cycle space

For an arbitrarily given numbering of the nodes is called an element incidence vector for the edge set if

Applies. Do the edges also a non-negative weight, the entries of the vector are multiplied by this weight. The set of all cycles as described form the cycle space, a subspace of. A basis of the cycle space are the fundamental cycles. Each fundamental circuit formed by adding an edge to a spanning tree.

The Kozyklenraum is the vector space of all incidence vectors generated by cuts. He is also a subspace of and results in direct sum with the cycle space of the room. A base of Kozyklenraums are the fundamental cuts. Each fundamental section is created by omitting an edge of a spanning tree as a connected component.

Cycle detection using depth-first search

V For each node: visited (v ) = false, finished ( v) = false For each node v:    DFS (v ) DFS (v ):    if finished ( v)      return    if visited (v)      " Cycle found " and demolition    visited (v ) = true    for each successor w      DFS (w)    finished ( v) = true literature

  • R. Diestel: Graph Theory. 3rd edition. Springer, Heidelberg 2005. ISBN 3-540-67656-2
  • Basic concept (graph theory )
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