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Definition df-cycl 24917
Description: Define the set of all (simple) cycles (in an undirected graph).

According to Wikipedia ("Cycle (graph theory)", https://en.wikipedia.org/wiki/Cycle_(graph_theory), 3-Oct-2017): "A simple cycle may be defined either as a closed walk with no repetitions of vertices and edges allowed, other than the repetition of the starting and ending vertex,"

According to Bollobas: "If a walk W = x0 x1 ... x(l) is such that l >= 3, x0=x(l), and the vertices x(i), 0 < i < l, are distinct from each other and x0, then W is said to be a cycle.", see Definition of [Bollobas] p. 5.

However, since a walk consisting of distinct vertices (except the first and the last vertex) is a path, a cycle can be defined as path whose first and last vertices coincide. So a cycle is represented by the following sequence: p(0) e(f(1)) p(1) ... p(n-1) e(f(n)) p(n)=p(0). (Contributed by Alexander van der Vekens, 3-Oct-2017.)

Assertion
Ref Expression
df-cycl  |- Cycles  =  ( v  e.  _V , 
e  e.  _V  |->  {
<. f ,  p >.  |  ( f ( v Paths 
e ) p  /\  ( p `  0
)  =  ( p `
 ( # `  f
) ) ) } )
Distinct variable group:    v, e, f, p

Detailed syntax breakdown of Definition df-cycl
StepHypRef Expression
1 ccycl 24911 . 2  class Cycles
2 vv . . 3  setvar  v
3 ve . . 3  setvar  e
4 cvv 3058 . . 3  class  _V
5 vf . . . . . . 7  setvar  f
65cv 1404 . . . . . 6  class  f
7 vp . . . . . . 7  setvar  p
87cv 1404 . . . . . 6  class  p
92cv 1404 . . . . . . 7  class  v
103cv 1404 . . . . . . 7  class  e
11 cpath 24904 . . . . . . 7  class Paths
129, 10, 11co 6277 . . . . . 6  class  ( v Paths 
e )
136, 8, 12wbr 4394 . . . . 5  wff  f ( v Paths  e ) p
14 cc0 9521 . . . . . . 7  class  0
1514, 8cfv 5568 . . . . . 6  class  ( p `
 0 )
16 chash 12450 . . . . . . . 8  class  #
176, 16cfv 5568 . . . . . . 7  class  ( # `  f )
1817, 8cfv 5568 . . . . . 6  class  ( p `
 ( # `  f
) )
1915, 18wceq 1405 . . . . 5  wff  ( p `
 0 )  =  ( p `  ( # `
 f ) )
2013, 19wa 367 . . . 4  wff  ( f ( v Paths  e ) p  /\  ( p `
 0 )  =  ( p `  ( # `
 f ) ) )
2120, 5, 7copab 4451 . . 3  class  { <. f ,  p >.  |  ( f ( v Paths  e
) p  /\  (
p `  0 )  =  ( p `  ( # `  f ) ) ) }
222, 3, 4, 4, 21cmpt2 6279 . 2  class  ( v  e.  _V ,  e  e.  _V  |->  { <. f ,  p >.  |  ( f ( v Paths  e
) p  /\  (
p `  0 )  =  ( p `  ( # `  f ) ) ) } )
231, 22wceq 1405 1  wff Cycles  =  ( v  e.  _V , 
e  e.  _V  |->  {
<. f ,  p >.  |  ( f ( v Paths 
e ) p  /\  ( p `  0
)  =  ( p `
 ( # `  f
) ) ) } )
Colors of variables: wff setvar class
This definition is referenced by:  cycls  25027  cyclispth  25033  cycliscrct  25034  cyclnspth  25035
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