matroid

matroid

A

matroid

is a combinatorial

structure

whose

properties

imitate those of linearly independent subsets of a

vector space

. Notions such as rank and independence (of a subset) have a meaning for any matroid, as does the notion of duality.

1

Definitions of a matroid

A matroid permits several

equivalent

formal

definitions

: two definitions
in terms of a

rank function

, one in terms of

independent

subsets, and
several more. We discuss several definitions below.

First rank definition

Definition 1

A matroid consists of a set

E

and a

function

r

:

P

⁢

(

E

)

→

N

satisfying the axioms:

r1

For any

S

∈

𝒫

⁢

(

E

)

,

r

⁢

(

S

)

≤

|

S

|

.

r2

The function

r

is

order-preserving

.

r3

The function

r

satisfies

the submodular inequality. That is, for any

S

,

T

∈

𝒫

⁢

(

E

)

,

r

⁢

(

S

∪

T

)

+

r

⁢

(

S

∩

T

)

≤

r

⁢

(

S

)

+

r

⁢

(

T

)

.

In this situation,

r

is called the

rank function

of the matroid

(

E

,

r

)

. If every

singleton

of

E

has rank equal to 1, then

(

E

,

r

)

is called a normal matroid.

An

isomorphism

of matroids

(

E

,

r

)

→

(

F

,

s

)

consists of a

bijection

f

:

E

→

F

which

preserves

rank, that is, satisfies

s

⁢

(

f

⁢

(

A

)

)

=

r

⁢

(

A

)

for all

A

∈

𝒫

⁢

(

E

)

.

Second rank definition

Definition 2

A matroid consists of a set

E

and a function

r

:

P

⁢

(

E

)

→

N

satisfying the axioms:

q1

r

⁢

(

∅

)

=

0

.

q2

If

x

∈

E

and

S

∈

𝒫

⁢

(

E

)

, then

r

⁢

(

S

∪

{

x

}

)

-

r

⁢

(

S

)

is either 0 or 1.

q3

If

x

,

y

∈

E

and

S

∈

𝒫

⁢

(

E

)

, then

r

⁢

(

S

∪

{

x

}

)

=

r

⁢

(

S

∪

{

y

}

)

=

r

⁢

(

S

)

implies

r

⁢

(

S

∪

{

x

,

y

}

)

=

r

⁢

(

S

)

.

Independent set definition

Definition 3

A matroid is a pair

(

E

,

I

)

with

I

⊆

P

⁢

(

E

)

(called the

independent sets

of

E

) satisfying the axioms:

i1

The

empty set

is independent.

i2

Every subset of an independent set is independent.

i3

For any

U

⊆

E

, any two subsets of

U

which are maximal with respect to membership in

ℐ

have the same

cardinality

.

The matroid

(

E

,

I

)

is normal if every singleton in

E

is independent.

Base definition

Definition 4

A matroid is a pair

(

E

,

B

)

with

B

⊆

P

⁢

(

E

)

(called the

bases

of

E

) a subset of

P

⁢

(

E

)

satisfying the axioms:

b1

E

has at least one base.

b2

The

proper subsets

of a base are not bases.

b3

If

S

and

T

are bases and

x

∈

E

∖

S

, then for some

y

∈

E

∖

T

, the set

(

S

∪

{

x

}

)

∖

{

y

}

is a base.

The matroid

(

E

,

ℬ

)

is called normal if each singleton of

E

is

contained

in a base of

E

.

Closure definition

Definition 5

A matroid consists of a set

E

and a function

cl

:

P

⁢

(

E

)

→

P

⁢

(

E

)

, called the

closure operator

, satisfying the axioms:

cl1

Any subset of

E

is contained in its

closure

.

cl2

If

S

⊆

cl

⁡

(

T

)

then

cl

⁡

(

S

)

⊆

cl

⁡

(

T

)

.

cl3

If

x

is in the closure of

S

∪

{

y

}

but not that of

S

, then

y

is in the closure of

S

∪

{

x

}

.

The closure operator is sometimes also called the

span

mapping

of the matroid. In this case

cl

⁡

(

A

)

is called the span of

A

.

The matroid

(

E

,

cl

)

is normal if the empty set is its own closure.

Circuit definition

Definition 6

A matroid is a pair

(

E

,

C

)

with

C

⊂

P

⁢

(

E

)

(called the

circuits

of

E

) satisfying the axioms:

c1

The empty set is not a circuit.

c2

The

proper supersets

of a circuit are not circuits.

c3

If

x

is in the

intersection

of two distinct circuits

S

and

T

, then there is a circuit

U

⊆

S

∪

T

which does not contain

x

.

The matroid

(

E

,

𝒞

)

is normal if no singleton of

E

is a circuit.

Combinatorial optimization definition

There’s yet another definition of matroids from “

Combinatorial Optimization

” by Papadimitriou and Steiglitz. pp. 280-285.

It requires three definitions to generate their definition of matroids. These are more or less grabbed from the book above.

A

subset system

(E,g) is a

finite set

E with g a

collection

of subsets of E

closed under

inclusion

, meaning that if

A

∈

g

, and

B

⊆

A

, then

B

∈

g

.

Definition 2: The

“combinatorial optimization problem”

(nonstandard term used in book) is as follows. Let (E,g) be a subset system and weight w, a nonnegative real function on E. Find the subset in g with the largest total weight.

Definition 3: Let (E,g) be a subset system and w, a weight function defined as above to give the “combinatorial optimization problem”. The

greedy algorithm

for construction of a subset I in g is as follows. Start with I being the empty set. Take the next highest weight

element

, e in E (w(e) ¿= w(f) for all f in E). If the union of I and e is in g, then add element e to I. Repeat until you exhaust all elements of E.

Now we have the definition of a matroid.

Definition 4:Let M=(E,g) be a subset system. M is a “matroid” if the greedy algorithm correctly solves the ”combinatorial optimization problem” for any weight function associated with M.

2

Equivalence of the definitions

It would take several pages to spell out what is a circuit in terms of rank,
and likewise for each other possible pair of the alternative defining notions,
and then to prove that the various sets of axioms unambiguously define the
same structure.
So let me sketch just one example: the equivalence of Definitions 1 (on
rank) and 6 (on circuits).
Assume first the conditions in Definition 1.
Define a circuit as a

minimal

subset

A

of

𝒫

⁢

(

E

)

having the property

r

⁢

(

A

)

<

|

A

|

.
With a little effort, we verify the axioms (c1)-(c3).
Now assume (c1)-(c3), and let

r

⁢

(

A

)

be the largest
integer

n

such that

A

has a subset

B

for which

–

B

contains no element of

C

–

n

=

|

B

|

.

One now proves (r1)-(r3).
Next, one shows that if we define

C

in terms of

r

, and then another rank function

s

in terms of

C

, we end up with

s

=

r

.
The equivalence of (r*) and (c*) is easy enough as well.

3

Examples of matroids

Let

V

be a vector space over a field

k

, and let

E

be a finite subset of

V

.
For

S

⊂

E

, let

r

⁢

(

S

)

be the

dimension

of the

subspace

of

V

generated by

S

.
Then

(

E

,

r

)

is a matroid.
Such a matroid, or one isomorphic
to it, is said to be

representable

over

k

.
The matroid is normal iff

0

∉

E

.
There exist matroids which are not representable over any field.

The second example of a matroid comes from

graph theory

.
The following definition will be rather informal, partly because the
terminology of graph theory is not very well standardised.
For our present purpose, a graph consists
of a finite set

V

, whose elements are called vertices, plus a set

E

of
two-element subsets of

V

, called edges.
A circuit in the graph is a finite set of at least three edges which can
be arranged in a cycle:

{

a

,

b

}

,

{

b

,

c

}

,

…

⁢

{

y

,

z

}

,

{

z

,

a

}

such that the vertices

a

,

b

,

…

⁢

z

are distinct.
With circuits thus defined,

E

satisfies the axioms in Definition 6,
and is thus a matroid, and in fact a normal matroid.
(The definition is easily adjusted to permit graphs with loops, which
define non-normal matroids.)
Such a matroid, or one isomorphic to it, is called “graphic”.

Let

E

=

A

∪

B

be a finite set, where

A

and

B

are nonempty and

disjoint

.
Let

G

a subset of

A

×

B

.
We get a “

matching

” matroid on

E

as follows.
Each element of

E

defines a “line” which is a subset
(a row or column) of the set

A

×

B

.
Let us call the elements of

G

“points”.
For any

S

⊂

E

let

r

⁢

(

S

)

be the largest number

n

such that for some set of points

P

:

–

|

P

|

=

n

– No two points of

P

are on the same line

– Any point of

P

is on a line defined by an element of

S

.

One can prove (it is not

trivial

) that

r

is the rank function of
a matroid on

E

.
That matroid is normal iff every line contains at least one point.
Matching matroids participate in

combinatorics

, in

connection

with results on “transversals”, such as Hall’s

marriage
theorem

.

4

The dual of a matroid

Proposition

: Let

E

be a matroid and

r

its rank function.
Define a mapping

s

:

β

⁢

(

E

)

→

ℕ

by

s

⁢

(

A

)

=

|

A

|

-

r

⁢

(

E

)

+

r

⁢

(

E

-

A

)

.

Then the pair

(

E

,

s

)

is a matroid (called the dual of

(

E

,

r

)

.

We leave the proof as an exercise.
Also, it is easy to verify that the
dual of the dual is the original matroid.
A circuit in

(

E

,

s

)

is also
referred to as a cocircuit in

(

E

,

r

)

.
There is a notion of cobasis also, and cospan.

If the dual of

E

is graphic,

E

is called cographic.
This notion of
duality agrees with the notion of same name in the theory of

planar
graphs

(and likewise in linear algebra): given a

plane graph

, the dual
of its matroid is the matroid of the

dual graph

.
A matroid that is both
graphic and cographic is called planar, and various criteria for
planarity of a graph can be extended to matroids.
The notion of
orientability can also be extended from graphs to matroids.

5

Binary matroids

A matroid is said to be binary if it is representable over the field of
two elements.
There are several other (equivalent)

characterisations

of a binary
matroid

(

E

,

r

)

, such as:

– The

symmetric difference

of any family of circuits is the union of
a family of

pairwise disjoint

circuits.

– For any circuit

C

and cocircuit

D

, we have

|

C

∩

D

|

≡

0

(

mod

2

)

.

Any

graphic matroid

is binary.
The dual of a binary matroid is binary.

6

Miscellaneous

The definition of the

chromatic polynomial

of a graph,

χ

⁢

(

x

)

=

∑

F

⊂

E

(

-

1

)

|

F

|

⁢

x

r

⁢

(

E

)

-

r

⁢

(

F

)

,

extends without change to any matroid.
This

polynomial

has something to say about the decomposibility of
matroids into simpler ones.

Also on the topic of decomposibility, matroids have a sort of
structure theory, in terms of what are called minors and separators.
That theory, due to Tutte, goes by

induction

; roughly speaking,
it is an adaptation of the old

algorithms

for putting
a matrix into a canonical form.

Along the same lines are several

theorems

on “basis exchange”, such as
the following.
Let

E

be a matroid and let

A

=

{

a

1

,

…

,

a

n

}

B

=

{

b

1

,

…

,

b

n

}

be two (equipotent) bases of

E

.
There exists a

permutation

ψ

of the set

{

1

,

…

,

n

}

such that, for every

m

from

0

to

n

,

{

a

1

,

…

,

a

m

,

b

ψ

⁢

(

m

+

1

)

,

…

,

b

ψ

⁢

(

n

)

}

is a basis of

E

.

7

Further reading

A good textbook is:

James G. Oxley,

Matroid Theory

,
Oxford University Press, New York etc., 1992

plus the updates-and-errata file at Dr. Oxley’s

http://www.math.lsu.edu/ oxley

website.

The chromatic polynomial is not discussed in Oxley, but see e.g.

http://www.math.binghamton.edu/zaslav/Matroids

Zaslavski.

Title

matroid

Canonical

name

Matroid

Date of creation

2013-03-22 13:08:56

Last modified on

2013-03-22 13:08:56

Owner

mps (409)

Last modified by

mps (409)

Numerical id

16

Author

mps (409)

Entry type

Definition

Classification

msc 05B35

Synonym

independence structure

Related topic

ChromaticPolynomial

Related topic

DependenceRelation

Defines

submodular inequality

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