(closed) walk / trek / trail / path

(closed) walk / trek / trail / path

Graph theory

terminology is notoriously variable so the following definitions should be used with caution. In books, most authors define their usage at the beginning.

In a graph,

multigraph

or even

pseudograph

G

,

•

a

walk

of length

s

is formed by a

sequence

of

s

edges such
that any two successive edges in the sequence share a vertex (aka node).
The walk is also considered to include all the vertices (nodes)

incident

to those edges, making it a

subgraph

.

In the case of a

simple graph

(i.e. not a multigraph) it is also
possible to define the walk uniquely by the vertices it visits: a

walk

of length

s

is then a sequence of vertices

ν

0

,

ν

1

, …

ν

s

such that an edge

ν

i

⁢

ν

i

+

1

exists
for all

0

⩽

i

<

s

. Again the walk is considered to contain those
edges as well as the vertices.

•

A

trek

is a walk that does not backtrack, i.e. no two successive edges are the same.

For simple graphs this also implies

ν

i

≠

ν

i

+

2

for all

0

⩽

i

⩽

s

-

2

.

•

A

trail

is a walk where all edges are distinct, and

•

a

path

is one where all vertices are distinct.

The walk, etc. is said to

run from

ν

0

to

ν

s

, to

run between

them, to

connect

them etc. The term

trek

was introduced by Cameron

[

Cam94

]

who notes the lexicographic mnemonic

𝑝𝑎𝑡ℎ𝑠

⊂

𝑡𝑟𝑎𝑖𝑙𝑠

⊂

𝑡𝑟𝑒𝑘𝑠

⊂

𝑤𝑎𝑙𝑘𝑠

The other terms are fairly widespread, cf.

[

Wil02

]

, but

beware:

many authors call walks

paths

, and some then call paths

chains

. And when edges are called

arcs

, a trek of length

s

sometimes goes by the name

s

-arc

.

Note that for the purpose of defining connectivity any of these types of wanderings can be used; if a walk exists between vertices

μ

and

ν

then there also exists a path between them. And here we must allow

s

=

0

to make “are

connected

by a path” an

equivalence relation

on vertices (in order to define

connected components

as its

equivalence classes

).

•

A

closed walk

aka

circuit

of length

s

≠

0

is a walk where

ν

0

=

ν

s

,

•

a

closed trek

is a trek that’s closed in the same way, and

•

a

closed trail

likewise;

•

a

closed path

aka (

elementary

)

cycle

is like a path
(except that we only demand that

ν

i

for

0

⩽

i

<

s

are distinct)
and again closed (

ν

s

again coincides with

ν

0

).

Beware:

cycles are often called circuits

[

Cam94

]

; the distinction between circuits and cycles here follows Wilson

[

Wil02

]

. These closed wanderings are often called after their length:

s

-circuits

,

s

-cycles

.

The case

s

=

0

is excluded from these definitions;

1

-cycles are loops so imply a pseudograph;

2

-cycles are double edges implying multigraphs; so

3

is the minimum cycle length in a proper graph.

Note also that in trivalent aka

cubic graphs

a closed trail is automatically a closed path: it is impossible to visit a vertex (in via edge

a

, out via edge

b

say) and visit it again (in via

c

, out via

d

) without also revisiting an edge, because there are only three edges at each vertex.

•

An

open walk

,

open trek

,

open trail

is one that isn’t
closed.

•

An

open path

(sometimes

open chain

) is just a path as
defined above (because a closed path isn’t actually a path). Still,
the term is useful when you want to emphasise the contrast with a
closed path.

References

1

Cam94

Peter J. Cameron

,

Combinatorics

: topics, techniques, algorithms

Camb. Univ. Pr. 1994,

ISBN  0 521 45761 0

,

http://www.maths.qmul.ac.uk/ pjc/comb/http://www.maths.qmul.ac.uk/ pjc/comb/

(solutions, errata &c.)

Wil02

Robert A. Wilson

,

Graphs,

Colourings

and the Four-colour

Theorem

,

Oxford Univ. Pr. 2002,

ISBN  0 19 851062 4

(pbk),

http://www.maths.qmul.ac.uk/ raw/graph.htmlhttp://www.maths.qmul.ac.uk/ raw/graph.html

(errata &c.)

Title

(closed) walk / trek / trail / path

Canonical name

closedWalkTrekTrailPath

Date of creation

2013-03-22 15:09:50

Last modified on

2013-03-22 15:09:50

Owner

marijke (8873)

Last modified by

marijke (8873)

Numerical id

9

Author

marijke (8873)

Entry type

Definition

Classification

msc 05C38

Related topic

Path2

Related topic

ConnectedGraph

Related topic

KConnectedGraph

Related topic

Diameter3

Defines

walk

Defines

trek

Defines

trail

Defines

path

Defines

chain

Defines

circuit

Defines

cycle

Defines

closed walk

Defines

closed trek

Defines

closed trail

Defines

closed path

Defines

closed chain

Defines

open walk

Defines

open trek

Defines

open trail

Defines

open path

Defines

open chain

Defines

s

-arc

Defines

s

-cycle

Defines

elementary cycle

Defines

s

-circuit

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