Editing Cycle (graph theory) (section)

== Definitions ==
=== Circuit and cycle ===
* A '''circuit''' is a non-empty [[Path (graph theory)#Walk, trail, and path|trail]] in which the first and last vertices are equal (''closed trail'').{{sfn|Bender|Williamson|2010|p=164}}
: Let {{nowrap|1=''G'' = (''V'', ''E'', ''Φ'')}} be a graph. A circuit is a non-empty trail {{nowrap|(''e''<sub>1</sub>, ''e''<sub>2</sub>, ..., ''e''<sub>''n''</sub>)}} with a vertex sequence {{nowrap|(''v''<sub>1</sub>, ''v''<sub>2</sub>, ..., ''v''<sub>''n''</sub>, ''v''<sub>1</sub>)}}.
* A '''cycle''' or '''simple circuit''' is a circuit in which only the first and last vertices are equal.{{sfn|Bender|Williamson|2010|p=164}}
* ''n'' is called the '''length of the circuit''' resp. '''length of the cycle'''.

=== Directed circuit and directed cycle ===
* A '''directed circuit''' is a non-empty [[Path (graph theory)#Directed walk, directed trail, and directed path|directed trail]] in which the first and last vertices are equal (''closed directed trail'').{{sfn|Bender|Williamson|2010|p=164}}
: Let {{nowrap|1=''G'' = (''V'', ''E'', ''Φ'')}} be a directed graph. A directed circuit is a non-empty directed trail {{nowrap|(''e''<sub>1</sub>, ''e''<sub>2</sub>, ..., ''e''<sub>''n''</sub>)}} with a vertex sequence {{nowrap|(''v''<sub>1</sub>, ''v''<sub>2</sub>, ..., ''v''<sub>''n''</sub>, ''v''<sub>1</sub>)}}.
* A '''directed cycle''' or '''simple directed circuit''' is a directed circuit in which only the first and last vertices are equal.{{sfn|Bender|Williamson|2010|p=164}}
* ''n'' is called the '''length of the directed circuit''' resp. '''length of the directed cycle'''.