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Definition df-trail 28319
Description: Define the set of all Trails (in an undirected graph).

According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory), 3-Oct-2017): "A trail is a walk in which all edges are distinct.

According to Bollobas: "... walk is called a trail if all its edges are distinct.", see Definition of [Bollobas] p. 5.

Therefore, a trail can be represented by an injective mapping f from { 1 , ... , n } and a mapping p from { 0 , ... , n }, where f enumerates the (indices of the) different edges, and p enumerates the vertices. So the trail 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) (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.)

Assertion
Ref Expression
df-trail  |- Trails  =  ( v  e.  _V , 
e  e.  _V  |->  {
<. f ,  p >.  |  ( f ( v Walks 
e ) p  /\  Fun  `' f ) } )
Distinct variable group:    v, e, f, p

Detailed syntax breakdown of Definition df-trail
StepHypRef Expression
1 ctrail 28310 . 2  class Trails
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 cwalk 28309 . . . . . . 7  class Walks
129, 10, 11co 5874 . . . . . 6  class  ( v Walks 
e )
136, 8, 12wbr 4039 . . . . 5  wff  f ( v Walks  e ) p
146ccnv 4704 . . . . . 6  class  `' f
1514wfun 5265 . . . . 5  wff  Fun  `' f
1613, 15wa 358 . . . 4  wff  ( f ( v Walks  e ) p  /\  Fun  `' f )
1716, 5, 7copab 4092 . . 3  class  { <. f ,  p >.  |  ( f ( v Walks  e
) p  /\  Fun  `' f ) }
182, 3, 4, 4, 17cmpt2 5876 . 2  class  ( v  e.  _V ,  e  e.  _V  |->  { <. f ,  p >.  |  ( f ( v Walks  e
) p  /\  Fun  `' f ) } )
191, 18wceq 1632 1  wff Trails  =  ( v  e.  _V , 
e  e.  _V  |->  {
<. f ,  p >.  |  ( f ( v Walks 
e ) p  /\  Fun  `' f ) } )
Colors of variables: wff set class
This definition is referenced by:  trls  28334  trliswlk  28337
  Copyright terms: Public domain W3C validator