Editing Cycle (graph theory) (section)

{{Short description|Trail in which only the first and last vertices are equal.}}
[[File:Graph cycle.svg|thumb|A graph with edges colored to illustrate a [[Path (graph theory)#Walk, trail, and path|closed walk]], H–A–B–A–H, in green; a circuit which is a closed walk in which all edges are distinct, B–D–E–F–D–C–B, in blue; and a cycle which is a closed walk in which all vertices are distinct, H–D–G–H, in red.]]

In [[graph theory]], a '''cycle''' in a [[Graph (discrete mathematics)|graph]] is a non-empty [[Path (graph theory)#Walk, trail, and path|trail]] in which only the first and last [[Vertex (graph theory)|vertices]] are equal. A '''directed cycle''' in a [[directed graph]] is a non-empty [[Path (graph theory)#Directed walk, directed trail, and directed path|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 acyclic graph]]''. A [[connected graph]] without cycles is called a ''[[Tree (graph theory)|tree]]''.