graph homomorphism

graph homomorphism

We shall define what a

graph homomorphism

is between two

pseudographs

, so that a graph homomorphism between two

multigraphs

, or two graphs are special cases.

Recall that a pseudograph

G

is an ordered triple

(

V

,

E

,

i

)

, where

V

is a set called the

vertex set

of

G

,

E

is a set called the

edge set

of

G

, and

i

:

E

→

2

V

is a map (incidence map), such that for every

e

∈

E

,

1

≤

|

i

⁢

(

e

)

|

≤

2

.

Elements of

V

are called vertices, elements of

E

edges. The vertex/vertices associated with any edge

e

via the map

i

is/are called the

endpoint

(s) of

e

. If an edge

e

has only one endpoint, it is called a loop.

e

1

and

e

2

are said to be

parallel

if

i

⁢

(

e

1

)

=

i

⁢

(

e

2

)

.

As two examples, a multigraph is a pseudograph such that

|

i

⁢

(

e

)

|

=

2

for each edge

e

∈

E

(no loops allowed). A graph is a multigraph such that

i

is

one-to-one

(no parallel edges allowed).

Definition

. Given two pseudographs

G

1

=

(

V

1

,

E

1

,

i

1

)

and

G

2

=

(

V

2

,

E

2

,

i

2

)

, a

graph homomorphism

h

from

G

1

to

G

2

consists of two

functions

f

:

V

1

→

V

2

and

g

:

E

1

→

E

2

, such that

i

2

∘

g

=

f

*

∘

i

1

.

(1)

The function

f

*

:

2

V

1

→

2

V

2

is defined as

f

*

⁢

(

S

)

=

{

f

⁢

(

s

)

∣

s

∈

S

}

.

A

graph isomorphism

h

=

(

f

,

g

)

is a graph homomorphism such that both

f

and

g

are

bijections

. It is a

graph automorphism

if

G

1

=

G

2

.

Remarks

.

•

In case when

G

1

and

G

2

are graphs, a graph homomorphism may be defined in terms of a single function

f

:

V

1

→

V

2

satisfying the condition (*)

{

v

1

,

v

2

}

is an edge of

G

1

⟹

{

f

⁢

(

v

1

)

,

f

⁢

(

v

2

)

}

is an edge of

G

2

.

To see this, suppose

h

=

(

f

,

g

)

is a graph homomorphism from

G

1

to

G

2

. Then

f

*

⁢

i

1

⁢

(

e

)

=

f

*

⁢

(

{

v

1

,

v

2

}

)

=

{

f

⁢

(

v

1

)

,

f

⁢

(

v

2

)

}

=

i

2

⁢

(

g

⁢

(

e

)

)

. The last equation says that

f

⁢

(

v

1

)

and

f

⁢

(

v

2

)

are endpoints of

g

⁢

(

e

)

.

Conversely

, suppose

f

:

V

1

→

V

2

is a function sastisfying condition (*). For each

e

∈

E

1

with

end points

{

v

1

,

v

2

}

, let

e

′

∈

E

2

be the edge whose endpoints are

{

f

⁢

(

v

1

)

,

f

⁢

(

v

2

)

}

. There is only one such edge because

G

2

is a graph, having no parallel edges. Define

g

:

E

1

→

E

2

by

g

⁢

(

e

)

=

e

′

. Then

g

is a well-defined function which also

satisfies

Equation (1). So

h

:=

(

f

,

g

)

is a graph homomorphism

G

1

→

G

2

.

•

The definition of a graph homomorphism between pseudographs can be analogously applied to one between directed pseudographs. Since the incidence map

i

now maps each edge to an

ordered pair

of vertices, a graph homomorphism thus defined will respect the “direction” of each edge. For example,

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are two non-isomorphic

digraphs

whose underlying graphs are

isomorphic

. Note that one digraph is

strongly connected

, while the other is only weakly connected.

Title

graph homomorphism

Canonical name

GraphHomomorphism

Date of creation

2013-03-22 16:04:53

Last modified on

2013-03-22 16:04:53

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

11

Author

CWoo (3771)

Entry type

Definition

Classification

msc 05C75

Classification

msc 05C60

Synonym

graph automorphism

Related topic

GraphIsomorphism

Related topic

Pseudograph

Related topic

Multigraph

Related topic

Graph

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