labelled digraph

labelled digraph

A triple

(

V

,

E

,

w

)

is called a

labelled digraph

, if

(

V

,

E

)

is a

digraph

and

w

is an association of

elements

from some set

L

, the

labels

, to some of the edges and vertices of the digraph. In
other words,

w

is a

mapping

from a subset

A

⊆

V

∪

E

to

L

. Most often,

L

is a subset of the

real numbers

, in which case

(

V

,

E

,

w

)

is called a

weighted

digraph and its labels are
called

weights

. Typically, either

A

=

V

or

A

=

E

, in which case

(

V

,

E

,

w

)

is called either a

vertex-weighted digraph

or an

edge-weighted digraph

, respectively.

Application examples

We give two typical “real life” examples. The first features an
edge-weighted digraph, while the second requires the implementation of
a vertex-weighted digraph.

Railway network

A railway

network

consists of railway stations

connected

by rails. A
train needs a certain time (measured in minutes) to fare from one
station to another. In a formalisation,

V

is the set of train
stations,

E

the set of direct connexions between them and

w

:

E

→

ℕ

a weighting corresponding to the journey times, so

(

V

,

E

,

w

)

is an edge-weighted digraph. Although typically

(

V

,

E

)

is a

http://planetmath.org/node/1702

symmetric

digraph,

w

does not need to be symmetric: for example, the
journey from

a

to

b

might take longer than the return journey
because

b

is located on a mountain.

An important optimisation problem is the efficient determination of
the fastest way from one station to another. An even harder problem is
to find the fastest round trip (usually called a

tour

) via a
given number of stations. This is the

travelling salesman
problem

.

Dependency graph

A software bundle consists of a number of packages each of which is
either installed or not. An installed package occupies a certain
amount of bytes on a storage medium. Packages may depend on other
packages, that is installation of a package may require other packages
to be installed first, which in turn may require still other packages
and so forth. One is interested in the

complete

storage requirement
incurred by the installation of one package and all its dependencies.

In a formalisation, the packages are vertices of a digraph

(

V

,

E

)

, and an
edge

(

a

,

b

)

∈

E

means “

a

depends on

b

”. Such a digraph is typically
not symmetric. The weighting

w

:

V

→

ℕ

associates sizes to
packages. A subset

W

of

V

is

dependency-closed

, if for any

w

∈

W

, all dependencies of

w

are in

W

. Given a to-be-installed
package

v

, the storage requirement incurred by the installation of

v

and all its dependencies is the sum of the vertex weights of the
smallest dependency-closed subset of

V

containing

v

.

Title

labelled digraph

Canonical name

LabelledDigraph

Date of creation

2013-03-22 15:15:07

Last modified on

2013-03-22 15:15:07

Owner

GrafZahl (9234)

Last modified by

GrafZahl (9234)

Numerical id

4

Author

GrafZahl (9234)

Entry type

Definition

Classification

msc 05C20

Classification

msc 05C12

Classification

msc 05C78

Classification

msc 05C90

Defines

label

Defines

weighted digraph

Defines

weight

Defines

vertex-weighted digraph

Defines

edge-weighted digraph

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