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Definition df-crct 28322
Description: Define the set of all circuits (in an undirected graph).

According to Wikipedia ("Cycle (graph theory)", https://en.wikipedia.org/wiki/Cycle_(graph_theory), 3-Oct-2017): "A circuit can be a closed walk allowing repetitions of vertices but not edges;"; according to Wikipedia ("Glossary of graph theory terms", https://en.wikipedia.org/wiki/Glossary_of_graph_theory_terms, 3-Oct-2017): "A circuit may refer to ... a trail (a closed tour without repeated edges), ...".

Following Bollobas ("A trail whose endvertices coincide (a closed trail) is called a circuit.", see Definition of [Bollobas] p. 5.), a circuit is a closed trail without repeated edges. So the circuit is also represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n)=p(0) (Contributed by Alexander van der Vekens, 3-Oct-2017.)

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

Detailed syntax breakdown of Definition df-crct
StepHypRef Expression
1 ccrct 28316 . 2  class Circuits
2 vv . . 3  set  v
3 ve . . 3  set  e
4 cvv 2801 . . 3  class  _V
5 vf . . . . . . 7  set  f
65cv 1631 . . . . . 6  class  f
7 vp . . . . . . 7  set  p
87cv 1631 . . . . . 6  class  p
92cv 1631 . . . . . . 7  class  v
103cv 1631 . . . . . . 7  class  e
11 ctrail 28310 . . . . . . 7  class Trails
129, 10, 11co 5874 . . . . . 6  class  ( v Trails 
e )
136, 8, 12wbr 4039 . . . . 5  wff  f ( v Trails  e ) p
14 cc0 8753 . . . . . . 7  class  0
1514, 8cfv 5271 . . . . . 6  class  ( p `
 0 )
16 chash 11353 . . . . . . . 8  class  #
176, 16cfv 5271 . . . . . . 7  class  ( # `  f )
1817, 8cfv 5271 . . . . . 6  class  ( p `
 ( # `  f
) )
1915, 18wceq 1632 . . . . 5  wff  ( p `
 0 )  =  ( p `  ( # `
 f ) )
2013, 19wa 358 . . . 4  wff  ( f ( v Trails  e ) p  /\  ( p `
 0 )  =  ( p `  ( # `
 f ) ) )
2120, 5, 7copab 4092 . . 3  class  { <. f ,  p >.  |  ( f ( v Trails  e
) p  /\  (
p `  0 )  =  ( p `  ( # `  f ) ) ) }
222, 3, 4, 4, 21cmpt2 5876 . 2  class  ( v  e.  _V ,  e  e.  _V  |->  { <. f ,  p >.  |  ( f ( v Trails  e
) p  /\  (
p `  0 )  =  ( p `  ( # `  f ) ) ) } )
231, 22wceq 1632 1  wff Circuits  =  ( v  e.  _V , 
e  e.  _V  |->  {
<. f ,  p >.  |  ( f ( v Trails 
e ) p  /\  ( p `  0
)  =  ( p `
 ( # `  f
) ) ) } )
Colors of variables: wff set class
This definition is referenced by:  crcts  28366  crctistrl  28372
  Copyright terms: Public domain W3C validator