presentations and extensions of transversal matroidshome.gwu.edu/~jbonin/presentations.pdflet r =...

Presentations and Extensions

of Transversal Matroids

Joseph E. Bonin

The George Washington University

This includes joint work withAnna de Mier (Universitat Politecnica de Catalunya)

The motivation, by analogy

A matroid M can have inequivalent representations, so a matrixrepresentation A can have extraneous information that limits whichextensions of M can be represented by adding a column to A.

aa′

bb′

cc′

aa′

bb′

cc′

.

The motivation, by analogy

A matroid M can have inequivalent representations, so a matrixrepresentation A can have extraneous information that limits whichextensions of M can be represented by adding a column to A.

aa′

bb′

cc′

aa′

bb′

cc′

.

Transversals matroid often have many presentations.

How do the presentations relate to (e.g., limit) the transversalextensions?

A brief review of transversal matroids

A set system, A = (Ai : i ∈ [r ]), is a multiset of sets.E.g., A1 = {a, b, d , e, f , g}, A2 = {b, c , f , g}, A3 = {d , e, f , g , }.

A1 = {a, b, d, e, f, g}

A2 = {b, c, f, g}

A3 = {d, e, f, g, }

a

b

c

d

e

f

g


A set system, A = (Ai : i ∈ [r ]), is a multiset of sets.E.g., A1 = {a, b, d , e, f , g}, A2 = {b, c , f , g}, A3 = {d , e, f , g , }.

A1 = {a, b, d, e, f, g}

A2 = {b, c, f, g}

A3 = {d, e, f, g, }

a

b

c

d

e

f

g

A1 = {a, b, d, e, f, g}

A2 = {b, c, f, g}

A3 = {d, e, f, g}

a

b

c

d

e

f

g

The independent sets of the transversal matroid M[A] are thepartial transversals of A.

The set system A is a presentation of M[A].


A1 = {a, b, d , e, f , g}, A2 = {b, c , f , g}, A3 = {d , e, f , g , }

A1

A2

A3

a b c d e f g

∗ ∗ 0 ∗ ∗ ∗ ∗0 ∗ ∗ 0 0 ∗ ∗0 0 0 ∗ ∗ ∗ ∗


A1 = {a, b, d , e, f , g}, A2 = {b, c , f , g}, A3 = {d , e, f , g , }

A1

A2

A3

a b c d e f g

∗ ∗ 0 ∗ ∗ ∗ ∗0 ∗ ∗ 0 0 ∗ ∗0 0 0 ∗ ∗ ∗ ∗

A1 = {a, b, d, e, f, g}

A2 = {b, c, f, g} A3 = {d, e, f, g, }

ce

a

bd

f g

We can replace ∗s with positive reals so that (i) determinants thataren’t forced to be zero for generic reasons are nonzero and(ii) each non-zero column sums to 1.

The multitude of presentations

4 of this type 1 of this type 8 of this type

4 of this type 2 of this type

1 of this type

The order on presentations

ce

a

fg

bd

A′1= {a, b, d, e, g}

A2 = {b, c, f, g} A3 = {d, e, f, g}

ce

a

fg

bd

{a, b, d, e, f, g}

{b, c, f, g} {d, e, f, g}

c e

a

fg

b d

{a, b, c, d, e, g}

{b, c, f, g} {d, e, f, g}

c e

a

f

gb d

A1 = {a, b, c, d, e, f, g}

A2 = {b, c, f, g} A3 = {d, e, f, g}

A partial order on thepresentations of M:

(A1, . . . ,Ar ) ≤ (A′1, . . . ,A

′r )

if, up to re-indexing the sets,Ai ⊆ A′

i for all i ∈ [r ].

The order on presentations

ce

a

fg

bd

A′1= {a, b, d, e, g}

A2 = {b, c, f, g} A3 = {d, e, f, g}

ce

a

fg

bd

{a, b, d, e, f, g}

{b, c, f, g} {d, e, f, g}

c e

a

fg

b d

{a, b, c, d, e, g}

{b, c, f, g} {d, e, f, g}

c e

a

f

gb d

A1 = {a, b, c, d, e, f, g}

A2 = {b, c, f, g} A3 = {d, e, f, g}

A partial order on thepresentations of M:

(A1, . . . ,Ar ) ≤ (A′1, . . . ,A

′r )

if, up to re-indexing the sets,Ai ⊆ A′

i for all i ∈ [r ].

A transversal matroid has aunique maximal presentation.

(Mason, 1971.)

Typically there are manyminimal presentations.

Extensions of a presentation of a transversal matroid

We consider only(a) presentations of M with r = r(M) sets, and(b) single-element transversal extensions of M of rank r .

Let A = (Ai : i ∈ [r ]) be a presentation of M.

For x 6∈ E (M) and J ⊆ [r ], set AJ = (AJi : i ∈ [r ]) where

AJi =

{

Ai ∪ x , if i ∈ J,Ai , otherwise.

The extension AJ of A yields the extension M[AJ ] of M.

Examples of extensions

A1

A2 A3 M [A∅]

x a loop

M [A{1}] M [A{1,2}]

M [A{1,3}] M [A{1,2,3}]

M[A{2}] = M[A{1,2}]

M[A{3}] = M[A{1,3}]

M[A{2,3}] = M[A{1,2,3}]

New results on the number of extensions of M from one presentation

8 extensions 8 extensions 8 extensions

6 extensions 5 extensions

5 extensions Let TA be the set oftransversal extensions of Mobtained by extending A.

TheoremLet r = r(M).

A is minimal iff |TA| = 2r ;otherwise |TA| ≤

34 · 2r .

If A neither is minimal norcovers a minimalpresentation, then|TA| ≤

58 · 2r . (2014)

A lattice

A1

A2 A3 M [A∅]

x a loop

M [A{1}] M [A{1,2}]

M [A{1,3}] M [A{1,2,3}]

∅

{1}

{1, 2} {1, 3}

{1, 2, 3}

M[A{2}] = M[A{1,2}]

M[A{3}] = M[A{1,3}]

M[A{2,3}] = M[A{1,2,3}]

A second example of such a lattice

a

b

c

f

d

e

gh

iA1 = {a, b, c} A4 = {g, h, i}

A3 = {d, e, f, g, h, i}

A2 = {b, c, d, e, f}

∅

{1} {2} {3}

{1, 2} {1, 3} {2, 3} {3, 4}

{1, 2, 3} {1, 3, 4} {2, 3, 4}

{1, 2, 3, 4}

Closure operators

A closure operator on a set S is a map σ : 2S → 2S for which

1. X ⊆ σ(X ) for all X ⊆ S ,

2. if X ⊆ Y ⊆ S , then σ(X ) ⊆ σ(Y ), and

3. σ(σ(X )) = σ(X ) for all X ⊆ S .

A σ-closed set is a subset X of S with σ(X ) = X .

The σ-closed sets, ordered by inclusion, form a lattice;the lattice operations are X ∨ Y = σ(X ∪ Y ) and X ∧ Y = X ∩ Y .

A closure operator related to extensions

Let A = (Ai : i ∈ [r ]) be a presentation of M, where r = r(M).

LemmaFor e ∈ E (M)− Ai , (A1,A2, . . . ,Ai−1,Ai ∪ e,Ai+1, . . . ,Ar )is a presentation of M iff e is a coloop of M\Ai . (Bondy & Welsh, 1971)




So, for J ⊆ [r ], there is a maximum K ⊆ [r ] with M[AJ ] = M[AK ].

Define σA : 2[r ] → 2[r ] by σA(J) = K .




So, for J ⊆ [r ], there is a maximum K ⊆ [r ] with M[AJ ] = M[AK ].

Define σA : 2[r ] → 2[r ] by σA(J) = K .

TheoremThe map σA is a closure operator on [r ]. The join of I and J inthe lattice LA of σA-closed sets is I ∪ J, so LA is distributive.If B = (Bi : i ∈ [r ]) is a presentation of M with Ai ⊆ Bi for all i ,then LB is a sublattice of LA. (2014)

LA and TA are isomorphic lattices

Order TA, the set of extensions of M obtained by extending A, bythe weak order.

For matroids M and N on E , we have M ≤w N in theweak order if rM(X ) ≤ rN(X ) for all X ⊆ E ;equivalently, every independent set of M is independent in N.

TheoremThe map J 7→ M[AJ ] is an isomorphism from LA onto TA,so TA is a distributive lattice. (2014)

Minimal presentations are most important for extensions

TheoremIf A ≤ B, then TB ⊆ TA.

If Ai ⊆ Bi for all i ∈ [r ], and if J ⊆ [r ], then M[AσB(J)] = M[BJ ].(2014)

Corollary

For any single-element transversal extension M ′ of M, there is aminimal presentation A of M and a set J with M ′ = M[AJ ]. (2013)

A characterization of minimal presentations

TheoremLet r = r(M). A presentation A of M is minimal iff |TA| = 2r .

(2013)



(2013)

LemmaThe presentation A is minimal iff each set in A is a cocircuit of M.

(Las Vergnas, 1971; Bondy and Welsh, 1971)



(2013)



If A is not minimal, then r(M\Ai ) < r − 1 for some Ai ∈ A.

Set J = [r ]− i .

So x is a coloop of M[AJ ]\Ai , so M[AJ ] = M[AJ∪i ], so |TA| < 2r .



(2013)



If A is minimal and i ∈ [r ], then r(M\Ai ) = r − 1.

If J ⊆ [r ] and i 6∈ J, then x is not a coloop of M[AJ ]\Ai

since M[AJ ]\Ai extends M\Ai , so J is closed. Thus, LA = 2[r ].

Non-minimal presentations lose at least a quarter of the extensions

TheoremIf A is not minimal, then |TA| = |LA| ≤

34 · 2r . (2014)

This follows by finding the maximal proper sublattices of 2[r ] thatcontain ∅ and [r ], which are

Lij ={

X ⊆ [r ] : X ∩ {i , j} 6= {i}}

for any distinct i , j in [r ]; also, |Lij | =34 · 2

r .

Non-minimal presentations lose at least a quarter of the extensions

TheoremIf A is not minimal, then |TA| = |LA| ≤

34 · 2r . (2014)

This follows by finding the maximal proper sublattices of 2[r ] thatcontain ∅ and [r ], which are

Lij ={

X ⊆ [r ] : X ∩ {i , j} 6= {i}}

for any distinct i , j in [r ]; also, |Lij | =34 · 2

r .

Finding the next-largest sublattices (two types), and showing thatthe number of extensions goes down, gives:

TheoremIf A neither is minimal nor covers a minimal presentation, then|TA| = |LA| ≤

58 · 2

r . (2014)

All sublattices of 2[r ] can be LA

TheoremLet L be a sublattice of 2[r ] (so I ∩ J ∈ L and I ∪ J ∈ L for allI , J ∈ L) with both ∅ and [r ] in L.

There is a transversal matroid M of rank r with L = LA, where Ais the maximal presentation of M.

For each integer n ≥ r , there is a presentation A of the uniformmatroid Ur ,n on [n] with L = LA. (2014)

How many extensions can be common to two presentations?

TheoremIf A and B are different presentations of M, then

2 ≤ |TA ∩ TB| ≤3

4· 2r . (2014)



2 ≤ |TA ∩ TB| ≤3

4· 2r . (2014)

TheoremThe set LA,B = {J ∈ LA : M[AJ ] = M[BK ] for some K ∈ LB}is a sublattice of LA. (2014)



2 ≤ |TA ∩ TB| ≤3

4· 2r . (2014)

TheoremThe set LA,B = {J ∈ LA : M[AJ ] = M[BK ] for some K ∈ LB}is a sublattice of LA. (2014)

Closure under unions follows from:

TheoremFor all subsets J and K of [r ], the join of M[AJ ] and M[AK ] in thelattice of all extensions of M is transversal and is M[AJ∪K ]. (2013)

The corresponding statement for meets is false.

A question

Under the weak order, the set of all single-element extensions of amatroid M is a lattice.

Question

Is the set of all single-element transversal extensions of M, orderedby the weak order, a lattice?

As we saw, TA is a lattice, but this question concerns the union ofTA over all presentations A of M.

a a′

b

b′ c

c′

M

.

a a′

b

b′ c

c′

M1

a a′

b

b′ c

c′

M2

a

a′b

b′ c

c′

extension meeta a′

b

b′ c

c′

transversal meet

loop

A question

Under the weak order, the set of all single-element extensions of amatroid M is a lattice.

Question

Is the set of all single-element transversal extensions of M, orderedby the weak order, a lattice?

As we saw, TA is a lattice, but this question concerns the union ofTA over all presentations A of M.

M

a a′

c

b b′

d

Mc

a a′

cx

b b′

d

Md

a a′

c

b b′

d x

extension join

a a′

c

b b′

dx

transversal join

a a′

c

b b′

d

x

Thank you for listening.

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