anna de mier - upc universitat politècnica de catalunya · transversal matroids anna de mier...
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![Page 1: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/1.jpg)
Transversal matroids
Anna de Mier
Universitat Politecnica de Catalunya
Barcelona, Spain
Let’s Matroid CUSO doctoral school, Neuchatel, March 25–27, 2015
![Page 2: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/2.jpg)
What’s ahead
I Definition, first properties, and examples
I Further properties of transversal matroids
I Characterizations of transversal matroids
I Other topics
![Page 3: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/3.jpg)
Definition, first properties, and examples
Keywords: set systems, partial transversals, transversal matroids,matrix representations, lattice path matroids, bicircular matroids
![Page 4: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/4.jpg)
Transversals of set systemsA set system is a collection A of subsets of a set S :
A = (Aj : j ∈ J)
(both S and J finite)
Ex S = [9], A = (A,B,C ,D) :
A = {1 2 6 9},B = {2 3 4 5 6 7 9},C = {5 6 8 9},D = {7 8 9}
Set systems are in bijection with bipartite graphs with stable partsJ and S
A CB D
4 5 6 7 8 92 31
![Page 5: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/5.jpg)
Transversals of set systemsA set system is a collection A of subsets of a set S :
A = (Aj : j ∈ J)
(both S and J finite)
Ex S = [9], A = (A,B,C ,D) :
A = {1 2 6 9},B = {2 3 4 5 6 7 9},C = {5 6 8 9},D = {7 8 9}
Set systems are in bijection with bipartite graphs with stable partsJ and S
A CB D
4 5 6 7 8 92 31
![Page 6: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/6.jpg)
Transversals of set systems
Def A partial transversal of A = (Aj : j ∈ J) is a subset T of Ssuch that there is an injective map ϕ : T → J such that
t ∈ Aϕ(t) for all t ∈ T
If |T | = |J|, partial transversals are called just transversals (akasystems of distinct representatives)
Ex
{6, 7, 8} is a partial transversal{5, 6, 7, 8} is a transversal
![Page 7: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/7.jpg)
Transversals of set systems
Def A partial transversal of A = (Aj : j ∈ J) is a subset T of Ssuch that there is an injective map ϕ : T → J such that
t ∈ Aϕ(t) for all t ∈ T
If |T | = |J|, partial transversals are called just transversals (akasystems of distinct representatives)Ex
A = {1 2 6 9},B = {2 3 4 5 6 7 9},C = {5 6 8 9},D = {7 8 9}
{6, 7, 8} is a partial transversal
{5, 6, 7, 8} is a transversal
![Page 8: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/8.jpg)
Transversals of set systems
Def A partial transversal of A = (Aj : j ∈ J) is a subset T of Ssuch that there is an injective map ϕ : T → J such that
t ∈ Aϕ(t) for all t ∈ T
If |T | = |J|, partial transversals are called just transversals (akasystems of distinct representatives)Ex
A = {1 2 6 9},B = {2 3 4 5 6 7 9},C = {5 6 8 9},D = {7 8 9}
{6, 7, 8} is a partial transversal
{5, 6, 7, 8} is a transversal
![Page 9: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/9.jpg)
Transversals of set systems
Def A partial transversal of A = (Aj : j ∈ J) is a subset T of Ssuch that there is an injective map ϕ : T → J such that
t ∈ Aϕ(t) for all t ∈ T
If |T | = |J|, partial transversals are called just transversals (akasystems of distinct representatives)Ex
A = {1 2 6 9},B = {2 3 4 5 6 7 9},C = {5 6 8 9},D = {7 8 9}
{6, 7, 8} is a partial transversal{5, 6, 7, 8} is a transversal
![Page 10: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/10.jpg)
Transversals of set systems
Def A partial transversal of A = (Aj : j ∈ J) is a subset T of Ssuch that there is an injective map ϕ : T → J such that
t ∈ Aϕ(t) for all t ∈ T
If |T | = |J|, partial transversals are called just transversals (akasystems of distinct representatives)Ex
A = {1 2 6 9},B = {2 3 4 5 6 7 9},C = {5 6 8 9},D = {7 8 9}
{6, 7, 8} is a partial transversal{5, 6, 7, 8} is a transversal
![Page 11: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/11.jpg)
Transversals of set systems
In terms of the associated bipartite graph, partial transversals arethe end-vertices in S of partial matchings of the graph
A CB D
4 5 6 7 8 92 31
{2, 5, 8, 9} is a partial transversal that corresponds to severalmatchings
![Page 12: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/12.jpg)
Transversals of set systems
In terms of the associated bipartite graph, partial transversals arethe end-vertices in S of partial matchings of the graph
A CB D
4 5 6 7 8 92 31
{2, 5, 8, 9} is a partial transversal
that corresponds to severalmatchings
![Page 13: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/13.jpg)
Transversals of set systems
In terms of the associated bipartite graph, partial transversals arethe end-vertices in S of partial matchings of the graph
A CB D
4 5 6 7 8 92 31
{2, 5, 8, 9} is a partial transversal that corresponds to severalmatchings
![Page 14: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/14.jpg)
Transversal matroids
Thm (Edmonds and Fulkerson 65)The partial transversals of a set system A are the independent setsof a matroid
This matroid is called a transversal matroid, denoted M[A]
The set system A is a presentation of the matroid
Proof: it will become transparent once we introduce another viewon set systems and transversals. But first let’s look at someexamples
![Page 15: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/15.jpg)
Transversal matroids
Thm (Edmonds and Fulkerson 65)The partial transversals of a set system A are the independent setsof a matroid
This matroid is called a transversal matroid, denoted M[A]
The set system A is a presentation of the matroid
Proof: it will become transparent once we introduce another viewon set systems and transversals. But first let’s look at someexamples
![Page 16: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/16.jpg)
Transversal matroids
Thm (Edmonds and Fulkerson 65)The partial transversals of a set system A are the independent setsof a matroid
This matroid is called a transversal matroid, denoted M[A]
The set system A is a presentation of the matroid
Proof: it will become transparent once we introduce another viewon set systems and transversals. But first let’s look at someexamples
![Page 17: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/17.jpg)
Transversal matroids: examples
I All uniform matroids Ur ,n are transversal
Take A = ([n], [n], (r). . ., [n])
I But many other presentations are possible
U3,6 = ({1 2 3 4}, {1 2 5 6}, {3 4 5 6})
I Which transversal matroid does the following set system give?
({1 2 4}, {2 3}, {4 5})
A rank-3 matroid with 2 lines with a common point, aka thecycle matroid of K4 minus an edge
I Transversal matroids are closed under deletion
![Page 18: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/18.jpg)
Transversal matroids: examples
I All uniform matroids Ur ,n are transversal
Take A = ([n], [n], (r). . ., [n])
I But many other presentations are possible
U3,6 = ({1 2 3 4}, {1 2 5 6}, {3 4 5 6})
I Which transversal matroid does the following set system give?
({1 2 4}, {2 3}, {4 5})
A rank-3 matroid with 2 lines with a common point, aka thecycle matroid of K4 minus an edge
I Transversal matroids are closed under deletion
![Page 19: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/19.jpg)
Transversal matroids: examples
I All uniform matroids Ur ,n are transversal
Take A = ([n], [n], (r). . ., [n])
I But many other presentations are possible
U3,6 = ({1 2 3 4}, {1 2 5 6}, {3 4 5 6})
I Which transversal matroid does the following set system give?
({1 2 4}, {2 3}, {4 5})
A rank-3 matroid with 2 lines with a common point, aka thecycle matroid of K4 minus an edge
I Transversal matroids are closed under deletion
![Page 20: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/20.jpg)
Transversal matroids: examples
I All uniform matroids Ur ,n are transversal
Take A = ([n], [n], (r). . ., [n])
I But many other presentations are possible
U3,6 = ({1 2 3 4}, {1 2 5 6}, {3 4 5 6})
I Which transversal matroid does the following set system give?
({1 2 4}, {2 3}, {4 5})
A rank-3 matroid with 2 lines with a common point, aka thecycle matroid of K4 minus an edge
I Transversal matroids are closed under deletion
![Page 21: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/21.jpg)
Transversal matroids: examples
I All uniform matroids Ur ,n are transversal
Take A = ([n], [n], (r). . ., [n])
I But many other presentations are possible
U3,6 = ({1 2 3 4}, {1 2 5 6}, {3 4 5 6})
I Which transversal matroid does the following set system give?
({1 2 4}, {2 3}, {4 5})
A rank-3 matroid with 2 lines with a common point, aka thecycle matroid of K4 minus an edge
I Transversal matroids are closed under deletion
![Page 22: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/22.jpg)
Transversal matroids: examples
I All uniform matroids Ur ,n are transversal
Take A = ([n], [n], (r). . ., [n])
I But many other presentations are possible
U3,6 = ({1 2 3 4}, {1 2 5 6}, {3 4 5 6})
I Which transversal matroid does the following set system give?
({1 2 4}, {2 3}, {4 5})
A rank-3 matroid with 2 lines with a common point, aka thecycle matroid of K4 minus an edge
I Transversal matroids are closed under deletion
![Page 23: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/23.jpg)
The matrix perspective
Given a set system (Aj , j ∈ J) on S consider a |J| × |S | matrixwith entries
mi ,j =
{0 if j 6∈ Ai
xij if j ∈ Ai
where the xij are algebraically independent numbers
Ex A = {1 2 6 9},B = {2 3 4 5 6 7 9},C = {5 6 8 9},D = {7 8 9}x11 x12 0 0 0 x16 0 0 x190 x22 x23 x24 x25 x26 x27 0 x290 0 0 0 x35 x36 0 x38 x390 0 0 0 0 0 x47 x48 x49
Which square submatrices have non-zero determinant?
![Page 24: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/24.jpg)
The matrix perspective
Given a set system (Aj , j ∈ J) on S consider a |J| × |S | matrixwith entries
mi ,j =
{0 if j 6∈ Ai
xij if j ∈ Ai
where the xij are algebraically independent numbers
Ex A = {1 2 6 9},B = {2 3 4 5 6 7 9},C = {5 6 8 9},D = {7 8 9}x11 x12 0 0 0 x16 0 0 x190 x22 x23 x24 x25 x26 x27 0 x290 0 0 0 x35 x36 0 x38 x390 0 0 0 0 0 x47 x48 x49
Which square submatrices have non-zero determinant?
![Page 25: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/25.jpg)
The matrix perspective
Which square submatrices have non-zero determinant?∣∣∣∣∣∣∣∣0 0 0 x19
x25 x27 0 x29x35 0 x38 x390 x47 x48 x49
∣∣∣∣∣∣∣∣ =
− x25x47x38x19 − x35x27x48x19 6= 0
I For T ⊆ S and ϕ : T → J, observe
t ∈ Aϕ(t) for all t ∈ T⇔∏t∈T
xϕ(t),t 6= 0
I As all of the xij are algebraically independent,T is a partial transversal ⇔ the columns corresponding to T
are linearly independent
![Page 26: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/26.jpg)
The matrix perspective
Which square submatrices have non-zero determinant?∣∣∣∣∣∣∣∣0 0 0 x19
x25 x27 0 x29x35 0 x38 x390 x47 x48 x49
∣∣∣∣∣∣∣∣ = − x25x47x38x19 − x35x27x48x19 6= 0
I For T ⊆ S and ϕ : T → J, observe
t ∈ Aϕ(t) for all t ∈ T⇔∏t∈T
xϕ(t),t 6= 0
I As all of the xij are algebraically independent,T is a partial transversal ⇔ the columns corresponding to T
are linearly independent
![Page 27: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/27.jpg)
The matrix perspective
Which square submatrices have non-zero determinant?∣∣∣∣∣∣∣∣0 0 0 x19
x25 x27 0 x29x35 0 x38 x390 x47 x48 x49
∣∣∣∣∣∣∣∣ = − x25x47x38x19 − x35x27x48x19 6= 0
I For T ⊆ S and ϕ : T → J, observe
t ∈ Aϕ(t) for all t ∈ T⇔∏t∈T
xϕ(t),t 6= 0
I As all of the xij are algebraically independent,T is a partial transversal ⇔ the columns corresponding to T
are linearly independent
![Page 28: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/28.jpg)
The matrix perspective
Which square submatrices have non-zero determinant?∣∣∣∣∣∣∣∣0 0 0 x19
x25 x27 0 x29x35 0 x38 x390 x47 x48 x49
∣∣∣∣∣∣∣∣ = − x25x47x38x19 − x35x27x48x19 6= 0
I For T ⊆ S and ϕ : T → J, observe
t ∈ Aϕ(t) for all t ∈ T⇔∏t∈T
xϕ(t),t 6= 0
I As all of the xij are algebraically independent,T is a partial transversal ⇔ the columns corresponding to T
are linearly independent
![Page 29: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/29.jpg)
Transversal matroids are representable
Thus, the partial transversals of a set systems are indeed theindependent sets of a matroid, and this matroid is representable
A stronger result is
Thm (Piff and Welsh 70)Every transversal matroid is representable over all sufficiently largefields
![Page 30: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/30.jpg)
Transversal matroids are representable
Thus, the partial transversals of a set systems are indeed theindependent sets of a matroid, and this matroid is representable
A stronger result is
Thm (Piff and Welsh 70)Every transversal matroid is representable over all sufficiently largefields
![Page 31: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/31.jpg)
The rank of a transversal matroid
Suppose s1, . . . , sr are a basis of M = M[A], and that si ∈ Ai
The matrix looks like(X Y
Z 0
)where det(X ) 6= 0
Thus, the first r rows span the row-space, so M = M[(A1, . . . ,Ar )]
ThmA rank-r transversal matroid always has a presentation with r setsMorevoer, if there are no coloops, all presentations have exactly rsets
![Page 32: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/32.jpg)
The rank of a transversal matroid
Suppose s1, . . . , sr are a basis of M = M[A], and that si ∈ Ai
The matrix looks like(X Y
Z 0
)where det(X ) 6= 0
Thus, the first r rows span the row-space, so M = M[(A1, . . . ,Ar )]
ThmA rank-r transversal matroid always has a presentation with r setsMorevoer, if there are no coloops, all presentations have exactly rsets
![Page 33: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/33.jpg)
The rank of a transversal matroid
Suppose s1, . . . , sr are a basis of M = M[A], and that si ∈ Ai
The matrix looks like(X Y
Z 0
)where det(X ) 6= 0
Thus, the first r rows span the row-space, so M = M[(A1, . . . ,Ar )]
ThmA rank-r transversal matroid always has a presentation with r setsMorevoer, if there are no coloops, all presentations have exactly rsets
![Page 34: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/34.jpg)
The rank of a transversal matroid
Suppose s1, . . . , sr are a basis of M = M[A], and that si ∈ Ai
The matrix looks like(X Y
Z 0
)where det(X ) 6= 0
Thus, the first r rows span the row-space, so M = M[(A1, . . . ,Ar )]
ThmA rank-r transversal matroid always has a presentation with r sets
Morevoer, if there are no coloops, all presentations have exactly rsets
![Page 35: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/35.jpg)
The rank of a transversal matroid
Suppose s1, . . . , sr are a basis of M = M[A], and that si ∈ Ai
The matrix looks like(X Y
Z 0
)where det(X ) 6= 0
Thus, the first r rows span the row-space, so M = M[(A1, . . . ,Ar )]
ThmA rank-r transversal matroid always has a presentation with r setsMorevoer, if there are no coloops, all presentations have exactly rsets
![Page 36: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/36.jpg)
Example: lattice path matroids
Lattice paths start at the origin and have steps E = (1, 0) andN = (0, 1)
We label each possible N step by its distance to the originConsider two non-crossing paths P and Q with common endpoints
![Page 37: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/37.jpg)
Example: lattice path matroids
Lattice paths start at the origin and have steps E = (1, 0) andN = (0, 1)
We label each possible N step by its distance to the origin
Consider two non-crossing paths P and Q with common endpoints
![Page 38: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/38.jpg)
Example: lattice path matroids
Lattice paths start at the origin and have steps E = (1, 0) andN = (0, 1)
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We label each possible N step by its distance to the origin
Consider two non-crossing paths P and Q with common endpoints
![Page 39: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/39.jpg)
Example: lattice path matroids
Lattice paths start at the origin and have steps E = (1, 0) andN = (0, 1)
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We label each possible N step by its distance to the originConsider two non-crossing paths P and Q with common endpoints
![Page 40: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/40.jpg)
Example: lattice path matroids
Lattice paths start at the origin and have steps E = (1, 0) andN = (0, 1)
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We label each possible N step by its distance to the originConsider two non-crossing paths P and Q with common endpoints
![Page 41: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/41.jpg)
Example: lattice path matroids
Let P be the set of lattice paths in the region bounded by P and Q
P
Q
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N2 = {2 3 4 5 6}
Each such path is determined by the labels of its N stepsLet Ni be the set of possible N steps at level iEach path in P gives a transversal of (N1, . . . ,Nr )
![Page 42: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/42.jpg)
Example: lattice path matroids
Let P be the set of lattice paths in the region bounded by P and Q
P
Q
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N2 = {2 3 4 5 6}
Each such path is determined by the labels of its N stepsLet Ni be the set of possible N steps at level iEach path in P gives a transversal of (N1, . . . ,Nr )
![Page 43: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/43.jpg)
Example: lattice path matroids
Let P be the set of lattice paths in the region bounded by P and Q
P
Q
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N2 = {2 3 4 5 6}
Each such path is determined by the labels of its N steps
Let Ni be the set of possible N steps at level iEach path in P gives a transversal of (N1, . . . ,Nr )
![Page 44: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/44.jpg)
Example: lattice path matroids
Let P be the set of lattice paths in the region bounded by P and Q
P
Q
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7 8 9
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N2 = {2 3 4 5 6}
Each such path is determined by the labels of its N stepsLet Ni be the set of possible N steps at level i
Each path in P gives a transversal of (N1, . . . ,Nr )
![Page 45: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/45.jpg)
Example: lattice path matroids
Let P be the set of lattice paths in the region bounded by P and Q
P
Q
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N4 = {7 8 9}N3 = {4 5 6 7}N2 = {2 3 4 5 6}N1 = {1 2 3}
Each such path is determined by the labels of its N stepsLet Ni be the set of possible N steps at level i
Each path in P gives a transversal of (N1, . . . ,Nr )
![Page 46: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/46.jpg)
Example: lattice path matroids
Let P be the set of lattice paths in the region bounded by P and Q
P
Q
2
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6 7
7 8 9
6
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N4 = {7 8 9}N3 = {4 5 6 7}N2 = {2 3 4 5 6}N1 = {1 2 3}
Each such path is determined by the labels of its N stepsLet Ni be the set of possible N steps at level iEach path in P gives a transversal of (N1, . . . ,Nr )
![Page 47: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/47.jpg)
Example: lattice path matroidsThm (Bonin, de Mier and Noy 2003)Let P and Q be two non-crossing lattice paths ending at (m, r)and consider the transversal matroid M[P,Q] with presentation(N1, . . . ,Nr )Then M[P,Q] has rank r , m + r elements and its bases are inbijection with lattice paths in P
Proof idea:
P
Q
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7 8 9
6
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A lattice path matroid is a matroid isomorphic to M[P,Q] forsome P,Q
![Page 48: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/48.jpg)
Example: lattice path matroidsThm (Bonin, de Mier and Noy 2003)Let P and Q be two non-crossing lattice paths ending at (m, r)and consider the transversal matroid M[P,Q] with presentation(N1, . . . ,Nr )Then M[P,Q] has rank r , m + r elements and its bases are inbijection with lattice paths in P
Proof idea:
P
Q
2
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3
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6 7
7 8 9
6
1
A lattice path matroid is a matroid isomorphic to M[P,Q] forsome P,Q
![Page 49: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/49.jpg)
Example: lattice path matroidsThm (Bonin, de Mier and Noy 2003)Let P and Q be two non-crossing lattice paths ending at (m, r)and consider the transversal matroid M[P,Q] with presentation(N1, . . . ,Nr )Then M[P,Q] has rank r , m + r elements and its bases are inbijection with lattice paths in P
Proof idea:
P
Q
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6 7
7 8 9
6
1
A lattice path matroid is a matroid isomorphic to M[P,Q] forsome P,Q
![Page 50: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/50.jpg)
Example: lattice path matroidsThm (Bonin, de Mier and Noy 2003)Let P and Q be two non-crossing lattice paths ending at (m, r)and consider the transversal matroid M[P,Q] with presentation(N1, . . . ,Nr )Then M[P,Q] has rank r , m + r elements and its bases are inbijection with lattice paths in P
Proof idea:
P
Q
2
2
3
3
4
4
5
5
6 7
7 8 9
6
1
A lattice path matroid is a matroid isomorphic to M[P,Q] forsome P,Q
![Page 51: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/51.jpg)
Example: lattice path matroidsThm (Bonin, de Mier and Noy 2003)Let P and Q be two non-crossing lattice paths ending at (m, r)and consider the transversal matroid M[P,Q] with presentation(N1, . . . ,Nr )Then M[P,Q] has rank r , m + r elements and its bases are inbijection with lattice paths in P
Proof idea:
P
Q
2
2
3
3
4
4
5
5
6 7
7 8 9
6
1
A lattice path matroid is a matroid isomorphic to M[P,Q] forsome P,Q
![Page 52: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/52.jpg)
Example: lattice path matroidsThm (Bonin, de Mier and Noy 2003)Let P and Q be two non-crossing lattice paths ending at (m, r)and consider the transversal matroid M[P,Q] with presentation(N1, . . . ,Nr )Then M[P,Q] has rank r , m + r elements and its bases are inbijection with lattice paths in P
Proof idea:
P
Q
2
2
3
3
4
4
5
5
6 7
7 8 9
6
1
A lattice path matroid is a matroid isomorphic to M[P,Q] forsome P,Q
![Page 53: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/53.jpg)
Example: lattice path matroidsThm (Bonin, de Mier and Noy 2003)Let P and Q be two non-crossing lattice paths ending at (m, r)and consider the transversal matroid M[P,Q] with presentation(N1, . . . ,Nr )Then M[P,Q] has rank r , m + r elements and its bases are inbijection with lattice paths in P
Proof idea:
P
Q
2
2
3
3
4
4
5
5
6 7
7 8 9
6
1
A lattice path matroid is a matroid isomorphic to M[P,Q] forsome P,Q
![Page 54: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/54.jpg)
Example: lattice path matroids
Can different pairs of paths give isomorphic lattice path matroids?
Turning the diagram 180 degrees around gives isomorphic matroids
Thm (Bonin and de Mier 06)The case described above is the only situation where differentpaths yield isomorphic lattice path matroids
![Page 55: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/55.jpg)
Example: lattice path matroids
Can different pairs of paths give isomorphic lattice path matroids?
Turning the diagram 180 degrees around gives isomorphic matroids
Thm (Bonin and de Mier 06)The case described above is the only situation where differentpaths yield isomorphic lattice path matroids
![Page 56: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/56.jpg)
Example: lattice path matroids
Can different pairs of paths give isomorphic lattice path matroids?
Turning the diagram 180 degrees around gives isomorphic matroids
Thm (Bonin and de Mier 06)The case described above is the only situation where differentpaths yield isomorphic lattice path matroids
![Page 57: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/57.jpg)
Example: bicircular matroids
Let G = (V ,E ) be a graphFor v ∈ V , let Av = {e ∈ E : e is incident with v}
M[(Av : v ∈ V )] is a transversal matroid on E
Who are its independent sets?
Edge-sets spanning trees or unicyclic graphs are independent:
![Page 58: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/58.jpg)
Example: bicircular matroids
Let G = (V ,E ) be a graphFor v ∈ V , let Av = {e ∈ E : e is incident with v}M[(Av : v ∈ V )] is a transversal matroid on E
Who are its independent sets?
Edge-sets spanning trees or unicyclic graphs are independent:
![Page 59: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/59.jpg)
Example: bicircular matroids
Let G = (V ,E ) be a graphFor v ∈ V , let Av = {e ∈ E : e is incident with v}M[(Av : v ∈ V )] is a transversal matroid on E
Who are its independent sets?
Edge-sets spanning trees or unicyclic graphs are independent:
![Page 60: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/60.jpg)
Example: bicircular matroids
Let G = (V ,E ) be a graphFor v ∈ V , let Av = {e ∈ E : e is incident with v}M[(Av : v ∈ V )] is a transversal matroid on E
Who are its independent sets?
Edge-sets spanning trees or unicyclic graphs are independent:
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Example: bicircular matroids
Let G = (V ,E ) be a graphFor v ∈ V , let Av = {e ∈ E : e is incident with v}M[(Av : v ∈ V )] is a transversal matroid on E
Who are its independent sets?
Edge-sets spanning trees or unicyclic graphs are independent:
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Example: bicircular matroids
Let G = (V ,E ) be a graphFor v ∈ V , let Av = {e ∈ E : e is incident with v}M[(Av : v ∈ V )] is a transversal matroid on E
Who are its independent sets?
Edge-sets spanning trees or unicyclic graphs are independent:
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Example: bicircular matroids
Thm (Matthews 77, Simoes-Pereira 72)The independent sets of the matroid M[(Av : v ∈ V )] are theedge-sets all whose components are trees or unicyclic.
The circuits correspond to the bicycles of the graph
Note that bicircular matroids are loopless
![Page 64: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/64.jpg)
Example: bicircular matroids
Thm (Matthews 77, Simoes-Pereira 72)The independent sets of the matroid M[(Av : v ∈ V )] are theedge-sets all whose components are trees or unicyclic.The circuits correspond to the bicycles of the graph
Note that bicircular matroids are loopless
![Page 65: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/65.jpg)
Example: bicircular matroids
Thm (Matthews 77, Simoes-Pereira 72)The independent sets of the matroid M[(Av : v ∈ V )] are theedge-sets all whose components are trees or unicyclic.The circuits correspond to the bicycles of the graph
Note that bicircular matroids are loopless
![Page 66: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/66.jpg)
Further properties of transversal matroids
Keywords: geometric representation, cyclic flats, contraction,duality, maximal and minimal presentations
![Page 67: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/67.jpg)
Geometric representation of transversal matroids
Consider a matrix representation with non-negative entries andwhere all column sums are 1:
Ex A = {1 2 6 9},B = {2 3 4 5 6 7 9},C = {5 6 8 9},D = {7 8 9}1 x12 0 0 0 x16 0 0 x190 x22 1 1 x25 x26 x27 0 x290 0 0 0 x35 x36 0 x38 x390 0 0 0 0 0 x47 x48 x49
Each column gives a point that lies on an (r − 1)-dimensionalsimplexEach set represents one standard-basis vector
![Page 68: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/68.jpg)
Geometric representation of transversal matroids
Consider a matrix representation with non-negative entries andwhere all column sums are 1:
Ex A = {1 2 6 9},B = {2 3 4 5 6 7 9},C = {5 6 8 9},D = {7 8 9}1 x12 0 0 0 x16 0 0 x190 x22 1 1 x25 x26 x27 0 x290 0 0 0 x35 x36 0 x38 x390 0 0 0 0 0 x47 x48 x49
Each column gives a point that lies on an (r − 1)-dimensionalsimplex
Each set represents one standard-basis vector
![Page 69: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/69.jpg)
Geometric representation of transversal matroids
Consider a matrix representation with non-negative entries andwhere all column sums are 1:
Ex A = {1 2 6 9},B = {2 3 4 5 6 7 9},C = {5 6 8 9},D = {7 8 9}1 x12 0 0 0 x16 0 0 x190 x22 1 1 x25 x26 x27 0 x290 0 0 0 x35 x36 0 x38 x390 0 0 0 0 0 x47 x48 x49
Each column gives a point that lies on an (r − 1)-dimensionalsimplexEach set represents one standard-basis vector
![Page 70: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/70.jpg)
Geometric representation of transversal matroidsEx A = {1 2 6 9},B = {2 3 4 5 6 7 9},C = {5 6 8 9},D = {7 8 9}
1 x12 0 0 0 x16 0 0 x190 x22 1 1 x25 x26 x27 0 x290 0 0 0 x35 x36 0 x38 x390 0 0 0 0 0 x47 x48 x49
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Geometric representation of transversal matroids
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Independent sets in the matroid correspond to affinely independentsets of pointsAll affine relationships are dictated by the faces of the simplexPoints lie on the faces “as freely as possible”
![Page 72: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/72.jpg)
Geometric representation of transversal matroids
A flat of a matroid is cyclic if it is a union of circuits (i.e., it has nocoloops)
Where are cyclic flats in the simplex representation?
B
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D
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All rank-k cyclic flats lie on (k − 1)-dimensional faces of thesimplex
![Page 73: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/73.jpg)
Geometric representation of transversal matroids
A flat of a matroid is cyclic if it is a union of circuits (i.e., it has nocoloops)Where are cyclic flats in the simplex representation?
B
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D
C
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All rank-k cyclic flats lie on (k − 1)-dimensional faces of thesimplex
![Page 74: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/74.jpg)
Geometric representation of transversal matroids
A flat of a matroid is cyclic if it is a union of circuits (i.e., it has nocoloops)Where are cyclic flats in the simplex representation?
B
A
D
C
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All rank-k cyclic flats lie on (k − 1)-dimensional faces of thesimplex
![Page 75: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/75.jpg)
Geometric representation of transversal matroids
A flat of a matroid is cyclic if it is a union of circuits (i.e., it has nocoloops)Where are cyclic flats in the simplex representation?
B
A
D
C
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All rank-k cyclic flats lie on (k − 1)-dimensional faces of thesimplex
![Page 76: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/76.jpg)
Geometric representation of transversal matroids
A flat of a matroid is cyclic if it is a union of circuits (i.e., it has nocoloops)Where are cyclic flats in the simplex representation?
B
A
D
C
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All rank-k cyclic flats lie on (k − 1)-dimensional faces of thesimplex
![Page 77: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/77.jpg)
Geometric representation of transversal matroids
A flat of a matroid is cyclic if it is a union of circuits (i.e., it has nocoloops)Where are cyclic flats in the simplex representation?
B
A
D
C
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All rank-k cyclic flats lie on (k − 1)-dimensional faces of thesimplex
![Page 78: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/78.jpg)
Geometric representation of transversal matroids
A flat of a matroid is cyclic if it is a union of circuits (i.e., it has nocoloops)Where are cyclic flats in the simplex representation?
B
A
D
C
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All rank-k cyclic flats lie on (k − 1)-dimensional faces of thesimplex
![Page 79: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/79.jpg)
Geometric representation of transversal matroids
A flat of a matroid is cyclic if it is a union of circuits (i.e., it has nocoloops)Where are cyclic flats in the simplex representation?
B
A
D
C
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All rank-k cyclic flats lie on (k − 1)-dimensional faces of thesimplex
![Page 80: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/80.jpg)
Geometric representation of transversal matroids
Thm (Brylawski 75)A matroid is transversal if and only if it can be represented on asimplex in such a way that every rank-k flat is the set of points ona (k − 1)-dimensional face
(Observe that this bounds the number of cyclic flats transversalmatroids can have)
It is straightforward to go from the set system presentation to theaffine representation on the simplex:
- label vertices of the simplex by sets in the presentation
- place each element x in the span of the vertices correspondingto the sets that contain x
So, if we are given the simplex representation, to recover the setswe just need to look at the complements of the facets
![Page 81: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/81.jpg)
Geometric representation of transversal matroids
Thm (Brylawski 75)A matroid is transversal if and only if it can be represented on asimplex in such a way that every rank-k flat is the set of points ona (k − 1)-dimensional face
(Observe that this bounds the number of cyclic flats transversalmatroids can have)
It is straightforward to go from the set system presentation to theaffine representation on the simplex:
- label vertices of the simplex by sets in the presentation
- place each element x in the span of the vertices correspondingto the sets that contain x
So, if we are given the simplex representation, to recover the setswe just need to look at the complements of the facets
![Page 82: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/82.jpg)
Geometric representation of transversal matroids
Thm (Brylawski 75)A matroid is transversal if and only if it can be represented on asimplex in such a way that every rank-k flat is the set of points ona (k − 1)-dimensional face
(Observe that this bounds the number of cyclic flats transversalmatroids can have)
It is straightforward to go from the set system presentation to theaffine representation on the simplex:
- label vertices of the simplex by sets in the presentation
- place each element x in the span of the vertices correspondingto the sets that contain x
So, if we are given the simplex representation, to recover the setswe just need to look at the complements of the facets
![Page 83: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/83.jpg)
Contractions and duality
I Transversal matroids are not closed under contractions
M/e
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I Transversal matroids are not closed under duality
I Duals of transversal matroids are called cotransversal or strictgammoids. The smallest minor closed class containingtransversal matroids is the class of gammoids
![Page 84: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/84.jpg)
Contractions and duality
I Transversal matroids are not closed under contractions
M/e
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I Transversal matroids are not closed under duality
I Duals of transversal matroids are called cotransversal or strictgammoids. The smallest minor closed class containingtransversal matroids is the class of gammoids
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Contractions and duality
I Transversal matroids are not closed under contractions
M/e
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I Transversal matroids are not closed under duality
I Duals of transversal matroids are called cotransversal or strictgammoids. The smallest minor closed class containingtransversal matroids is the class of gammoids
![Page 86: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/86.jpg)
Presentations
Transversal matroids typically have several presentations
B
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A = {1 2 6 9},B = {2 3 4 5 6 7 9},C = {5 6 8 9},D = {7 8 9}
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Presentations
Transversal matroids typically have several presentations
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A = {1 2 6 9},B = {2 3 4 6 7 9},C = {5 6 8 9},D = {7 8 9}
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Presentations
Transversal matroids typically have several presentations
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A = {1 2 6 9},B = {2 3 4 5 6 7 9},C = {5 6 8 9},D = {7 8 9}
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Presentations
Transversal matroids typically have several presentations
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A = {1 2 6 9},B = {2 3 4 5 6 7},C = {5 6 8 9},D = {7 8 9}
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Presentations
Transversal matroids typically have several presentations
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A = {1 2 6 9},B = {2 3 4 5 6 7 9},C = {5 6 8 9},D = {7 8 9}
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Presentations
Transversal matroids typically have several presentations
B
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A = {1 2 6 9},B = {2 3 4 5 6 9},C = {5 6 8 9},D = {7 8 9}
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Presentations
Transversal matroids typically have several presentations
B
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A = {1 2 6 9},B = {2 3 4 5 6 8 9},C = {5 6 8 9},D = {7 8 9}
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Presentations
Transversal matroids typically have several presentations
B
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A = {1 2 6 9},B = {2 3 4 5 6 8},C = {5 6 8 9},D = {7 8 9}
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Presentations
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I Aci is a flat
I (Bondy and Welsh 71) a is a coloop of Ac1 if and only if
(A1 ∪ a,A2, . . . ,Ar ) is also a presentation
I (Mason 69, Bondy 72) a transversal matroid has a uniquemaximal presentation (i.e., one in which sets cannot beincreased)
I But there can be several minimal presentations
![Page 95: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/95.jpg)
Presentations
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I Aci is a flat
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I (Mason 69, Bondy 72) a transversal matroid has a uniquemaximal presentation (i.e., one in which sets cannot beincreased)
I But there can be several minimal presentations
![Page 96: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/96.jpg)
Presentations
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I (Mason 69, Bondy 72) a transversal matroid has a uniquemaximal presentation (i.e., one in which sets cannot beincreased)
I But there can be several minimal presentations
![Page 97: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/97.jpg)
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I Aci is a flat
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(A1 ∪ a,A2, . . . ,Ar ) is also a presentation
I (Mason 69, Bondy 72) a transversal matroid has a uniquemaximal presentation (i.e., one in which sets cannot beincreased)
I But there can be several minimal presentations
![Page 98: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/98.jpg)
Presentations
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I Aci is a flat
I (Bondy and Welsh 71) a is a coloop of Ac1 if and only if
(A1 ∪ a,A2, . . . ,Ar ) is also a presentation
I (Mason 69, Bondy 72) a transversal matroid has a uniquemaximal presentation (i.e., one in which sets cannot beincreased)
I But there can be several minimal presentations
![Page 99: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/99.jpg)
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7
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I Aci is a flat
I (Bondy and Welsh 71) a is a coloop of Ac1 if and only if
(A1 ∪ a,A2, . . . ,Ar ) is also a presentation
I (Mason 69, Bondy 72) a transversal matroid has a uniquemaximal presentation (i.e., one in which sets cannot beincreased)
I But there can be several minimal presentations
![Page 100: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/100.jpg)
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1
34
9
56
2
7
8
I Aci is a flat
I (Bondy and Welsh 71) a is a coloop of Ac1 if and only if
(A1 ∪ a,A2, . . . ,Ar ) is also a presentation
I (Mason 69, Bondy 72) a transversal matroid has a uniquemaximal presentation (i.e., one in which sets cannot beincreased)
I But there can be several minimal presentations
![Page 101: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/101.jpg)
Example: bicircular matroids
I Bicircular matroids are those transversal matroids for whichthe simplex representation has points only on vertices andedges
I Deletions of bicircular matroids are bicircular
I Contractions of bicircular matroids are bicircular, if loopless
I Bicircular matroids are not closed under dualityFor instance, U2,n is bicircular but Un−2,n is not if n ≥ 7
![Page 102: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/102.jpg)
Example: bicircular matroids
I Bicircular matroids are those transversal matroids for whichthe simplex representation has points only on vertices andedges
I Deletions of bicircular matroids are bicircular
I Contractions of bicircular matroids are bicircular, if loopless
I Bicircular matroids are not closed under dualityFor instance, U2,n is bicircular but Un−2,n is not if n ≥ 7
![Page 103: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/103.jpg)
Example: bicircular matroids
I Bicircular matroids are those transversal matroids for whichthe simplex representation has points only on vertices andedges
I Deletions of bicircular matroids are bicircular
I Contractions of bicircular matroids are bicircular, if loopless
I Bicircular matroids are not closed under dualityFor instance, U2,n is bicircular but Un−2,n is not if n ≥ 7
![Page 104: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/104.jpg)
Example: bicircular matroids
I Bicircular matroids are those transversal matroids for whichthe simplex representation has points only on vertices andedges
I Deletions of bicircular matroids are bicircular
I Contractions of bicircular matroids are bicircular, if loopless
I Bicircular matroids are not closed under dualityFor instance, U2,n is bicircular but Un−2,n is not if n ≥ 7
![Page 105: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/105.jpg)
Example: lattice path matroids
I Lattice path matroids are closed under duality
I Lattice path matroids are closed under deletion andcontraction
I The excluded minors for the minor-closed class of lattice pathmatroids are known (Bonin 10)
![Page 106: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/106.jpg)
Example: lattice path matroids
I Lattice path matroids are closed under duality
*
I Lattice path matroids are closed under deletion andcontraction
I The excluded minors for the minor-closed class of lattice pathmatroids are known (Bonin 10)
![Page 107: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/107.jpg)
Example: lattice path matroids
I Lattice path matroids are closed under duality
*
I Lattice path matroids are closed under deletion andcontraction
4
4
I The excluded minors for the minor-closed class of lattice pathmatroids are known (Bonin 10)
![Page 108: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/108.jpg)
Example: lattice path matroids
I Lattice path matroids are closed under duality
*
I Lattice path matroids are closed under deletion andcontraction
4
4
I The excluded minors for the minor-closed class of lattice pathmatroids are known (Bonin 10)
![Page 109: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/109.jpg)
Example: lattice path matroids
I Lattice path matroids are closed under duality
*
I Lattice path matroids are closed under deletion andcontraction
4
4
I The excluded minors for the minor-closed class of lattice pathmatroids are known (Bonin 10)
![Page 110: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/110.jpg)
Example: lattice path matroids
I Lattice path matroids are closed under duality
*
I Lattice path matroids are closed under deletion andcontraction
I The excluded minors for the minor-closed class of lattice pathmatroids are known (Bonin 10)
![Page 111: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/111.jpg)
Example: lattice path matroids
I Lattice path matroids are closed under duality
*
I Lattice path matroids are closed under deletion andcontraction
I The excluded minors for the minor-closed class of lattice pathmatroids are known (Bonin 10)
![Page 112: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/112.jpg)
Characterizations of transversal matroids
Keywords: fundamental transvesal matroids, Mason-Ingletoninequalities, the β function
![Page 113: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/113.jpg)
The Mason-Ingleton characterization
How can one tell if a given matroid is transversal?
Thm (Mason 71, Ingleton 77)The following are equivalent
- M is transversal
- for all non-empty collections F of cyclic flats
r
( ⋂F∈F
F
)≤∑F ′⊆F
(−1)|F′|+1r
( ⋃F ′∈F ′
F ′
)
![Page 114: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/114.jpg)
The Mason-Ingleton characterization
How can one tell if a given matroid is transversal?
Thm (Mason 71, Ingleton 77)The following are equivalent
- M is transversal
- for all non-empty collections F of cyclic flats
r
( ⋂F∈F
F
)≤∑F ′⊆F
(−1)|F′|+1r
( ⋃F ′∈F ′
F ′
)
![Page 115: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/115.jpg)
The Mason-Ingleton characterization
A transversal matroid M is called fundamental transversal if thereis a basis B = {b1, . . . , br} of M such that in some simplexrepresentation of M the elements b1, . . . , br are placed on vertices
Ex U3,6 is a fundamental transversal matroid
But this one is not
![Page 116: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/116.jpg)
The Mason-Ingleton characterization
A transversal matroid M is called fundamental transversal if thereis a basis B = {b1, . . . , br} of M such that in some simplexrepresentation of M the elements b1, . . . , br are placed on vertices
Ex U3,6 is a fundamental transversal matroid
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But this one is not
![Page 117: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/117.jpg)
The Mason-Ingleton characterization
A transversal matroid M is called fundamental transversal if thereis a basis B = {b1, . . . , br} of M such that in some simplexrepresentation of M the elements b1, . . . , br are placed on vertices
Ex U3,6 is a fundamental transversal matroid
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But this one is not
![Page 118: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/118.jpg)
The Mason-Ingleton characterization
A transversal matroid M is called fundamental transversal if thereis a basis B = {b1, . . . , br} of M such that in some simplexrepresentation of M the elements b1, . . . , br are placed on vertices
Ex U3,6 is a fundamental transversal matroid
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But this one is not
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![Page 119: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/119.jpg)
The Mason-Ingleton characterization
Recall that a rank-k cyclic flat always lies on a (k − 1)-dimensionalface of the simplex
Thus, if F is a cyclic flat of a fundamental transversal matroid,then
r(F ) = |B ∩ F |
And similarly, for a collection F of cyclic flats,
r
( ⋃F∈F
F
)=
∣∣∣∣∣B ∩ (⋃F∈F
F )
∣∣∣∣∣ =
∣∣∣∣∣ ⋃F∈F
B ∩ F
∣∣∣∣∣r
( ⋂F∈F
F
)=
∣∣∣∣∣B ∩ (⋂F∈F
F )
∣∣∣∣∣ =
∣∣∣∣∣ ⋂F∈F
B ∩ F
∣∣∣∣∣
![Page 120: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/120.jpg)
The Mason-Ingleton characterization
Recall that a rank-k cyclic flat always lies on a (k − 1)-dimensionalface of the simplex
Thus, if F is a cyclic flat of a fundamental transversal matroid,then
r(F ) = |B ∩ F |
And similarly, for a collection F of cyclic flats,
r
( ⋃F∈F
F
)=
∣∣∣∣∣B ∩ (⋃F∈F
F )
∣∣∣∣∣ =
∣∣∣∣∣ ⋃F∈F
B ∩ F
∣∣∣∣∣r
( ⋂F∈F
F
)=
∣∣∣∣∣B ∩ (⋂F∈F
F )
∣∣∣∣∣ =
∣∣∣∣∣ ⋂F∈F
B ∩ F
∣∣∣∣∣
![Page 121: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/121.jpg)
The Mason-Ingleton characterization
Thus, for a fundamental transversal matroid, the Principle ofInclusion and Exclusion gives
r( ⋂F∈F
F)
=∑F ′⊆F
(−1)|F′|+1r
( ⋃F ′∈F ′
F ′)
And if M is not fundamental transversal? We can extend M to afundamental transversal matroid M1 by adding new elementsa1, . . . , ar on vertices
For a cyclic flat F of M, the flat F1 = cl1(F ) is cyclic in M1 andr(F ) = r1(F1)
And for a collection F of cyclic flats of M,
r(⋃
F∈F F)
= r ′(⋃
F1∈F1F1
)r(⋂
F∈F F)≤ r ′
(⋂F1∈F1
F1
)
![Page 122: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/122.jpg)
The Mason-Ingleton characterization
Thus, for a fundamental transversal matroid, the Principle ofInclusion and Exclusion gives
r( ⋂F∈F
F)
=∑F ′⊆F
(−1)|F′|+1r
( ⋃F ′∈F ′
F ′)
And if M is not fundamental transversal?
We can extend M to afundamental transversal matroid M1 by adding new elementsa1, . . . , ar on vertices
For a cyclic flat F of M, the flat F1 = cl1(F ) is cyclic in M1 andr(F ) = r1(F1)
And for a collection F of cyclic flats of M,
r(⋃
F∈F F)
= r ′(⋃
F1∈F1F1
)r(⋂
F∈F F)≤ r ′
(⋂F1∈F1
F1
)
![Page 123: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/123.jpg)
The Mason-Ingleton characterization
Thus, for a fundamental transversal matroid, the Principle ofInclusion and Exclusion gives
r( ⋂F∈F
F)
=∑F ′⊆F
(−1)|F′|+1r
( ⋃F ′∈F ′
F ′)
And if M is not fundamental transversal? We can extend M to afundamental transversal matroid M1 by adding new elementsa1, . . . , ar on vertices
For a cyclic flat F of M, the flat F1 = cl1(F ) is cyclic in M1 andr(F ) = r1(F1)
And for a collection F of cyclic flats of M,
r(⋃
F∈F F)
= r ′(⋃
F1∈F1F1
)r(⋂
F∈F F)≤ r ′
(⋂F1∈F1
F1
)
![Page 124: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/124.jpg)
The Mason-Ingleton characterization
Thus, for a fundamental transversal matroid, the Principle ofInclusion and Exclusion gives
r( ⋂F∈F
F)
=∑F ′⊆F
(−1)|F′|+1r
( ⋃F ′∈F ′
F ′)
And if M is not fundamental transversal? We can extend M to afundamental transversal matroid M1 by adding new elementsa1, . . . , ar on vertices
For a cyclic flat F of M, the flat F1 = cl1(F ) is cyclic in M1 andr(F ) = r1(F1)
And for a collection F of cyclic flats of M,
r(⋃
F∈F F)
= r ′(⋃
F1∈F1F1
)r(⋂
F∈F F)≤ r ′
(⋂F1∈F1
F1
)
![Page 125: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/125.jpg)
The Mason-Ingleton characterization
Thus, for a fundamental transversal matroid, the Principle ofInclusion and Exclusion gives
r( ⋂F∈F
F)
=∑F ′⊆F
(−1)|F′|+1r
( ⋃F ′∈F ′
F ′)
And if M is not fundamental transversal? We can extend M to afundamental transversal matroid M1 by adding new elementsa1, . . . , ar on vertices
For a cyclic flat F of M, the flat F1 = cl1(F ) is cyclic in M1 andr(F ) = r1(F1)
And for a collection F of cyclic flats of M,
r(⋃
F∈F F)
= r ′(⋃
F1∈F1F1
)r(⋂
F∈F F)≤ r ′
(⋂F1∈F1
F1
)
![Page 126: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/126.jpg)
The Mason-Ingleton characterizationWe have proved the ⇓ implication of
Thm (Mason 71, Ingleton 77)The following are equivalent
- M is transversal
- for all non-empty collection F of cyclic flats
r( ⋂F∈F
F)≤∑F ′⊆F
(−1)|F′|+1r
( ⋃F ′∈F ′
F ′)
And also the ⇓ implication of
Thm (Bonin, Kung and de Mier 11)The following are equivalent
- M is fundamental transversal
- for all non-empty collection F of cyclic flats
r( ⋂F∈F
F)
=∑F ′⊆F
(−1)|F′|+1r
( ⋃F ′∈F ′
F ′)
![Page 127: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/127.jpg)
The Mason-Ingleton characterizationWe have proved the ⇓ implication of
Thm (Mason 71, Ingleton 77)The following are equivalent
- M is transversal
- for all non-empty collection F of cyclic flats
r( ⋂F∈F
F)≤∑F ′⊆F
(−1)|F′|+1r
( ⋃F ′∈F ′
F ′)
And also the ⇓ implication of
Thm (Bonin, Kung and de Mier 11)The following are equivalent
- M is fundamental transversal
- for all non-empty collection F of cyclic flats
r( ⋂F∈F
F)
=∑F ′⊆F
(−1)|F′|+1r
( ⋃F ′∈F ′
F ′)
![Page 128: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/128.jpg)
The β function
What about the implication ⇑ in Mason-Ingleton’s theorem?
Every transvesal matroid has a unique maximal presentation; inthis presentation, the flats Ac
i are actually cyclic flats
Let Z(M) be the set of all cyclic flats of M
Suppose we had a map β : Z(M)→ {0, 1, 2, . . .} such that β(F ) isthe number of times F c appears in the maximal presentation of M
Thus, ∑F∈Z
β(F ) = r(M)
Moreover, as each F must intersect r(F ) sets of the presentation,∑Y∈Z,F∩Y c 6=∅
β(Y ) = r(F )
![Page 129: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/129.jpg)
The β function
What about the implication ⇑ in Mason-Ingleton’s theorem?
Every transvesal matroid has a unique maximal presentation; inthis presentation, the flats Ac
i are actually cyclic flats
Let Z(M) be the set of all cyclic flats of M
Suppose we had a map β : Z(M)→ {0, 1, 2, . . .} such that β(F ) isthe number of times F c appears in the maximal presentation of M
Thus, ∑F∈Z
β(F ) = r(M)
Moreover, as each F must intersect r(F ) sets of the presentation,∑Y∈Z,F∩Y c 6=∅
β(Y ) = r(F )
![Page 130: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/130.jpg)
The β function
What about the implication ⇑ in Mason-Ingleton’s theorem?
Every transvesal matroid has a unique maximal presentation; inthis presentation, the flats Ac
i are actually cyclic flats
Let Z(M) be the set of all cyclic flats of M
Suppose we had a map β : Z(M)→ {0, 1, 2, . . .} such that β(F ) isthe number of times F c appears in the maximal presentation of M
Thus, ∑F∈Z
β(F ) = r(M)
Moreover, as each F must intersect r(F ) sets of the presentation,∑Y∈Z,F∩Y c 6=∅
β(Y ) = r(F )
![Page 131: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/131.jpg)
The β function
What about the implication ⇑ in Mason-Ingleton’s theorem?
Every transvesal matroid has a unique maximal presentation; inthis presentation, the flats Ac
i are actually cyclic flats
Let Z(M) be the set of all cyclic flats of M
Suppose we had a map β : Z(M)→ {0, 1, 2, . . .} such that β(F ) isthe number of times F c appears in the maximal presentation of M
Thus, ∑F∈Z
β(F ) = r(M)
Moreover, as each F must intersect r(F ) sets of the presentation,∑Y∈Z,F∩Y c 6=∅
β(Y ) = r(F )
![Page 132: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/132.jpg)
The β function
What about the implication ⇑ in Mason-Ingleton’s theorem?
Every transvesal matroid has a unique maximal presentation; inthis presentation, the flats Ac
i are actually cyclic flats
Let Z(M) be the set of all cyclic flats of M
Suppose we had a map β : Z(M)→ {0, 1, 2, . . .} such that β(F ) isthe number of times F c appears in the maximal presentation of M
Thus, ∑F∈Z
β(F ) = r(M)
Moreover, as each F must intersect r(F ) sets of the presentation,∑Y∈Z,F∩Y c 6=∅
β(Y ) = r(F )
![Page 133: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/133.jpg)
The β function
Substracting the previous two equations, the map β should satisfy∑Y∈Z,F⊆Y
β(Y ) = r(M)− r(F )
For any matroid M, define the map β : 2S → Z on all subsets ofelements as
β(X ) = r(M)− r(X )−∑
Y∈Z,X⊂Yβ(Y )
(Ingleton and Piff 72, Mason 71)
Lem If M satisfies the Mason-Ingleton inequalities, thenβ(X ) ≥ 0 for all X
![Page 134: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/134.jpg)
The β function
Substracting the previous two equations, the map β should satisfy∑Y∈Z,F⊆Y
β(Y ) = r(M)− r(F )
For any matroid M, define the map β : 2S → Z on all subsets ofelements as
β(X ) = r(M)− r(X )−∑
Y∈Z,X⊂Yβ(Y )
(Ingleton and Piff 72, Mason 71)
Lem If M satisfies the Mason-Ingleton inequalities, thenβ(X ) ≥ 0 for all X
![Page 135: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/135.jpg)
The β function
Substracting the previous two equations, the map β should satisfy∑Y∈Z,F⊆Y
β(Y ) = r(M)− r(F )
For any matroid M, define the map β : 2S → Z on all subsets ofelements as
β(X ) = r(M)− r(X )−∑
Y∈Z,X⊂Yβ(Y )
(Ingleton and Piff 72, Mason 71)
Lem If M satisfies the Mason-Ingleton inequalities, thenβ(X ) ≥ 0 for all X
![Page 136: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/136.jpg)
The β function
β(S) = 0, β(X ) = r(M)− r(X )−∑
Y∈Z(M) :X⊂Y β(Y )
��������
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a
fg
ed
b
c
β({a, b, c}) = 3− 2− 0 = 1
β({a, d , e}) = β({a, f , g}) = 1
β(∅) = 3− 0− (1 + 1 + 1 + 0) = 0
β({a}) = 3− 1− (1 + 1 + 1 + 0) = −1
![Page 137: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/137.jpg)
The β function
β(S) = 0, β(X ) = r(M)− r(X )−∑
Y∈Z(M) :X⊂Y β(Y )
��������
��������
��������
��
����
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a
fg
ed
b
c
β({a, b, c}) = 3− 2− 0 = 1
β({a, d , e}) = β({a, f , g}) = 1
β(∅) = 3− 0− (1 + 1 + 1 + 0) = 0
β({a}) = 3− 1− (1 + 1 + 1 + 0) = −1
![Page 138: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/138.jpg)
The β function
β(S) = 0, β(X ) = r(M)− r(X )−∑
Y∈Z(M) :X⊂Y β(Y )
��������
��������
��������
��
����
��������
a
fg
ed
b
c
β({a, b, c}) = 3− 2− 0 = 1
β({a, d , e}) = β({a, f , g}) = 1
β(∅) = 3− 0− (1 + 1 + 1 + 0) = 0
β({a}) = 3− 1− (1 + 1 + 1 + 0) = −1
![Page 139: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/139.jpg)
The β function
β(S) = 0, β(X ) = r(M)− r(X )−∑
Y∈Z(M) :X⊂Y β(Y )
��������
��������
��������
��
����
��������
a
fg
ed
b
c
β({a, b, c}) = 3− 2− 0 = 1
β({a, d , e}) = β({a, f , g}) = 1
β(∅) = 3− 0− (1 + 1 + 1 + 0) = 0
β({a}) = 3− 1− (1 + 1 + 1 + 0) = −1
![Page 140: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/140.jpg)
The β function
It is not very difficult to check that if M is a matroid withβ(X ) ≥ 0 for all X ⊆ S , then the set system
(F c : F ∈ Z, β(F ) > 0)
is a presentation of the matroid M
So all together:
Thm The following are equivalent
- M is transversal
- for all non-empty collection F of cyclic flats
r( ⋂F∈F
F)≤∑F ′⊆F
(−1)|F′|+1r
( ⋂F ′∈F ′
F ′)
- β(X ) ≥ 0 for all subsets X of the ground set
![Page 141: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/141.jpg)
The β function
It is not very difficult to check that if M is a matroid withβ(X ) ≥ 0 for all X ⊆ S , then the set system
(F c : F ∈ Z, β(F ) > 0)
is a presentation of the matroid M
So all together:
Thm The following are equivalent
- M is transversal
- for all non-empty collection F of cyclic flats
r( ⋂F∈F
F)≤∑F ′⊆F
(−1)|F′|+1r
( ⋂F ′∈F ′
F ′)
- β(X ) ≥ 0 for all subsets X of the ground set
![Page 142: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/142.jpg)
Other topics
Keywords: Tutte polynomials of lattice path matroids, transversalextensions, cyclic ordering conjecture
![Page 143: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/143.jpg)
Tutte polynomials of lattice path matroids
Thm (Colbourn, Provan and Vertigan 95)Computing t(M; x , y) is #P-complete for transversal matroids
Can we do better for lattice path matroids?
Thm (Bonin, de Mier and Noy 03)The Tutte polynomial of a lattice path matroid can be computedin polynomial time
![Page 144: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/144.jpg)
Tutte polynomials of lattice path matroids
Thm (Colbourn, Provan and Vertigan 95)Computing t(M; x , y) is #P-complete for transversal matroids
Can we do better for lattice path matroids?
Thm (Bonin, de Mier and Noy 03)The Tutte polynomial of a lattice path matroid can be computedin polynomial time
![Page 145: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/145.jpg)
Tutte polynomials of lattice path matroids
Recall:
t(M; x , y) =∑
B∈B(M)
x i(B)y e(B)
where i(B), e(B) are the numbers of internally/externally activeelements with respect to basis B and some linear order < on theground set
x 6∈ B is externally active ifwhenever B − b ∪ x is a basis we have x < b
b ∈ B is internally active ifwhenever B − b ∪ x is a basis we have b < x
If B is a basis of a lattice path matroid, can one easily tell whichelements are internally/externally active?
![Page 146: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/146.jpg)
Tutte polynomials of lattice path matroids
Recall:
t(M; x , y) =∑
B∈B(M)
x i(B)y e(B)
where i(B), e(B) are the numbers of internally/externally activeelements with respect to basis B and some linear order < on theground set
x 6∈ B is externally active ifwhenever B − b ∪ x is a basis we have x < b
b ∈ B is internally active ifwhenever B − b ∪ x is a basis we have b < x
If B is a basis of a lattice path matroid, can one easily tell whichelements are internally/externally active?
![Page 147: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/147.jpg)
Tutte polynomials of lattice path matroids
Let M be a LPM of rank r and ground set [m + r ]; take the linearorder 1 < 2 < · · ·Let B = {b1 < · · · < bn} be a basis of a lattice path matroid andx 6∈ B; then bi < x < bi+1 for some i
x
Can we modify the path to include x by removing one step fromb1, . . . , bi? If we cannot, x is externally activeWe can use x in place of bi , provided we are not in the lower path!
![Page 148: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/148.jpg)
Tutte polynomials of lattice path matroids
Let M be a LPM of rank r and ground set [m + r ]; take the linearorder 1 < 2 < · · ·Let B = {b1 < · · · < bn} be a basis of a lattice path matroid andx 6∈ B; then bi < x < bi+1 for some i
x
Can we modify the path to include x by removing one step fromb1, . . . , bi? If we cannot, x is externally active
We can use x in place of bi , provided we are not in the lower path!
![Page 149: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/149.jpg)
Tutte polynomials of lattice path matroids
Let M be a LPM of rank r and ground set [m + r ]; take the linearorder 1 < 2 < · · ·Let B = {b1 < · · · < bn} be a basis of a lattice path matroid andx 6∈ B; then bi < x < bi+1 for some i
x
Can we modify the path to include x by removing one step fromb1, . . . , bi? If we cannot, x is externally active
We can use x in place of bi , provided we are not in the lower path!
![Page 150: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/150.jpg)
Tutte polynomials of lattice path matroids
Let M be a LPM of rank r and ground set [m + r ]; take the linearorder 1 < 2 < · · ·Let B = {b1 < · · · < bn} be a basis of a lattice path matroid andx 6∈ B; then bi < x < bi+1 for some i
x
Can we modify the path to include x by removing one step fromb1, . . . , bi? If we cannot, x is externally active
We can use x in place of bi , provided we are not in the lower path!
![Page 151: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/151.jpg)
Tutte polynomials of lattice path matroids
Let M be a LPM of rank r and ground set [m + r ]; take the linearorder 1 < 2 < · · ·Let B = {b1 < · · · < bn} be a basis of a lattice path matroid andx 6∈ B; then bi < x < bi+1 for some i
x
Can we modify the path to include x by removing one step fromb1, . . . , bi? If we cannot, x is externally active
We can use x in place of bi , provided we are not in the lower path!
![Page 152: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/152.jpg)
Tutte polynomials of lattice path matroids
Let M be a LPM of rank r and ground set [m + r ]; take the linearorder 1 < 2 < · · ·Let B = {b1 < · · · < bn} be a basis of a lattice path matroid andx 6∈ B; then bi < x < bi+1 for some i
x
Can we modify the path to include x by removing one step fromb1, . . . , bi? If we cannot, x is externally activeWe can use x in place of bi
, provided we are not in the lower path!
![Page 153: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/153.jpg)
Tutte polynomials of lattice path matroids
Let M be a LPM of rank r and ground set [m + r ]; take the linearorder 1 < 2 < · · ·Let B = {b1 < · · · < bn} be a basis of a lattice path matroid andx 6∈ B; then bi < x < bi+1 for some i
x
Can we modify the path to include x by removing one step fromb1, . . . , bi? If we cannot, x is externally activeWe can use x in place of bi , provided we are not in the lower path!
![Page 154: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/154.jpg)
Tutte polynomials of lattice path matroids
Lem An element x 6∈ B is externally active if the E step of Bcorresponding to x belongs to the lower bounding pathDually, an element b ∈ B is internally active if the N step of Bcorresponding to b belongs to the upper bounding path
In other words,
t(M[P,Q]; x , y) =∑π∈P
xQN(π)yPE (π)
where PE (π)/QN(π) are the number of E /N steps of π incommon with P/Q
![Page 155: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/155.jpg)
Tutte polynomials of lattice path matroids
Lem An element x 6∈ B is externally active if the E step of Bcorresponding to x belongs to the lower bounding pathDually, an element b ∈ B is internally active if the N step of Bcorresponding to b belongs to the upper bounding path
In other words,
t(M[P,Q]; x , y) =∑π∈P
xQN(π)yPE (π)
where PE (π)/QN(π) are the number of E /N steps of π incommon with P/Q
![Page 156: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/156.jpg)
Tutte polynomials of lattice path matroids
A byproduct:
As turning the paths 180 degrees around is a matroid isomorphismthat switches the paths P and Q, and we should get the sameTutte polynomial no matter how we compute it, one gets that thepairs
(QN(π),PE (π)) and (PN(π),QE (π))
have the same distribution over the paths π ∈ P
(For more on similar results, see Elizalde and Rubey 13+)
![Page 157: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/157.jpg)
Tutte polynomials of lattice path matroids
A byproduct:
As turning the paths 180 degrees around is a matroid isomorphismthat switches the paths P and Q, and we should get the sameTutte polynomial no matter how we compute it, one gets that thepairs
(QN(π),PE (π)) and (PN(π),QE (π))
have the same distribution over the paths π ∈ P
(For more on similar results, see Elizalde and Rubey 13+)
![Page 158: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/158.jpg)
Tutte polynomials of lattice path matroids
Thm (Bonin, de Mier and Noy 03)The Tutte polynomial of a lattice path matroid can be computedin polynomial time
- Label point (0, 0) with 1
- Recursively, label each point in the region determined by thebounding paths according to
- The label of the point (m, r) is t(M[P,Q]; x , y)
![Page 159: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/159.jpg)
Tutte polynomials of lattice path matroids
Thm (Bonin, de Mier and Noy 03)The Tutte polynomial of a lattice path matroid can be computedin polynomial time
- Label point (0, 0) with 1
- Recursively, label each point in the region determined by thebounding paths according to
g
f+g
f yf f
f xf
- The label of the point (m, r) is t(M[P,Q]; x , y)
![Page 160: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/160.jpg)
Transversal extensions
A (single-element) extension of a matroid M on E is a matroid Non E ∪ x such that M = N\x (and, for us, r(N) = r(M))
I The theory of extensions is well-understood: extensions of Mare in bijection with some families of subsets called “modularcuts of flats” (Crapo 65)
A transversal extension of a transversal matroid M is an extensionof M that is also transversal
I If (B1, . . . ,Br ) is a presentation of N, then (B1\x , . . . ,Br\x)is a presentation of M = N\x
I So all transversal extensions of M can be obtained by addingx to some sets in some presentation of M.
I But could it be that we get repetitions? How can we ensurewe have all extensions?
![Page 161: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/161.jpg)
Transversal extensions
A (single-element) extension of a matroid M on E is a matroid Non E ∪ x such that M = N\x (and, for us, r(N) = r(M))
I The theory of extensions is well-understood: extensions of Mare in bijection with some families of subsets called “modularcuts of flats” (Crapo 65)
A transversal extension of a transversal matroid M is an extensionof M that is also transversal
I If (B1, . . . ,Br ) is a presentation of N, then (B1\x , . . . ,Br\x)is a presentation of M = N\x
I So all transversal extensions of M can be obtained by addingx to some sets in some presentation of M.
I But could it be that we get repetitions? How can we ensurewe have all extensions?
![Page 162: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/162.jpg)
Transversal extensions
A (single-element) extension of a matroid M on E is a matroid Non E ∪ x such that M = N\x (and, for us, r(N) = r(M))
I The theory of extensions is well-understood: extensions of Mare in bijection with some families of subsets called “modularcuts of flats” (Crapo 65)
A transversal extension of a transversal matroid M is an extensionof M that is also transversal
I If (B1, . . . ,Br ) is a presentation of N, then (B1\x , . . . ,Br\x)is a presentation of M = N\x
I So all transversal extensions of M can be obtained by addingx to some sets in some presentation of M.
I But could it be that we get repetitions? How can we ensurewe have all extensions?
![Page 163: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/163.jpg)
Transversal extensions
A (single-element) extension of a matroid M on E is a matroid Non E ∪ x such that M = N\x (and, for us, r(N) = r(M))
I The theory of extensions is well-understood: extensions of Mare in bijection with some families of subsets called “modularcuts of flats” (Crapo 65)
A transversal extension of a transversal matroid M is an extensionof M that is also transversal
I If (B1, . . . ,Br ) is a presentation of N, then (B1\x , . . . ,Br\x)is a presentation of M = N\x
I So all transversal extensions of M can be obtained by addingx to some sets in some presentation of M.
I But could it be that we get repetitions? How can we ensurewe have all extensions?
![Page 164: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/164.jpg)
Transversal extensions
A (single-element) extension of a matroid M on E is a matroid Non E ∪ x such that M = N\x (and, for us, r(N) = r(M))
I The theory of extensions is well-understood: extensions of Mare in bijection with some families of subsets called “modularcuts of flats” (Crapo 65)
A transversal extension of a transversal matroid M is an extensionof M that is also transversal
I If (B1, . . . ,Br ) is a presentation of N, then (B1\x , . . . ,Br\x)is a presentation of M = N\x
I So all transversal extensions of M can be obtained by addingx to some sets in some presentation of M.
I But could it be that we get repetitions? How can we ensurewe have all extensions?
![Page 165: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/165.jpg)
Transversal extensions
Let A = (A1, . . . ,Ar ) be a presentation of MFor I ⊆ [r ], let
AI =
{Ai ∪ x , if i ∈ I ,Ai , otherwise.
The matroid M[AI ] is a transversal extension of M
Example: A = ({1, 2, 5, 6, 7, 8}, {3, 4, 5, 6, 7}, {7, 8})
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Transversal extensionsLet A = (A1, . . . ,Ar ) be a presentation of MFor I ⊆ [r ], let
AI =
{Ai ∪ x , if i ∈ I ,Ai , otherwise.
The matroid M[AI ] is a transversal extension of M
Example: A = ({1, 2, 5, 6, 7, 8}, {3, 4, 5, 6, 7}, {7, 8})
I = {1, 2, 3}
v3v
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Transversal extensionsLet A = (A1, . . . ,Ar ) be a presentation of MFor I ⊆ [r ], let
AI =
{Ai ∪ x , if i ∈ I ,Ai , otherwise.
The matroid M[AI ] is a transversal extension of M
Example: A = ({1, 2, 5, 6, 7, 8}, {3, 4, 5, 6, 7}, {7, 8})
I = {1, 2, 3}
v3v
2
v11
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Transversal extensionsLet A = (A1, . . . ,Ar ) be a presentation of MFor I ⊆ [r ], let
AI =
{Ai ∪ x , if i ∈ I ,Ai , otherwise.
The matroid M[AI ] is a transversal extension of M
Example: A = ({1, 2, 5, 6, 7, 8}, {3, 4, 5, 6, 7}, {7, 8})
I = {2, 3}
v3v
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v11
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Transversal extensionsLet A = (A1, . . . ,Ar ) be a presentation of MFor I ⊆ [r ], let
AI =
{Ai ∪ x , if i ∈ I ,Ai , otherwise.
The matroid M[AI ] is a transversal extension of M
Example: A = ({1, 2, 5, 6, 7, 8}, {3, 4, 5, 6, 7}, {7, 8})
I = {3}
v3v
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Transversal extensionsLet A = (A1, . . . ,Ar ) be a presentation of MFor I ⊆ [r ], let
AI =
{Ai ∪ x , if i ∈ I ,Ai , otherwise.
The matroid M[AI ] is a transversal extension of M
Example: A = ({1, 2, 5, 6, 7, 8}, {3, 4, 5, 6, 7}, {7, 8})
I = {1, 2}
v3v
2
v11
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4
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Transversal extensions
Some results from Bonin and de Mier 15
Thm The following are equivalent:
(i) if I 6= J then M[AI ] 6= M[AJ ]
(ii) the presentation A is minimal
Cor If A is a minimal presentation of M, then AI is a minimalpresentation of M[AI ]
Thm If N is a transversal extension of M, there exist a minimalpresentation A of M and a set I ⊆ [r ] such that N = M[AI ]
![Page 172: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/172.jpg)
Transversal extensions
Some results from Bonin and de Mier 15
Thm The following are equivalent:
(i) if I 6= J then M[AI ] 6= M[AJ ]
(ii) the presentation A is minimal
Cor If A is a minimal presentation of M, then AI is a minimalpresentation of M[AI ]
Thm If N is a transversal extension of M, there exist a minimalpresentation A of M and a set I ⊆ [r ] such that N = M[AI ]
![Page 173: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/173.jpg)
Transversal extensions
Some results from Bonin and de Mier 15
Thm The following are equivalent:
(i) if I 6= J then M[AI ] 6= M[AJ ]
(ii) the presentation A is minimal
Cor If A is a minimal presentation of M, then AI is a minimalpresentation of M[AI ]
Thm If N is a transversal extension of M, there exist a minimalpresentation A of M and a set I ⊆ [r ] such that N = M[AI ]
![Page 174: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/174.jpg)
Transversal extensions
Let M1,M2 be two matroids on E . The weak order:
M1 ≤w M2 if every independent set in M1 is independent in M2
Fact: the set of all extensions of a matroid M is a lattice under theweak order
Question: is the set of all transversal extensions of a transversalmatroid also a lattice under the weak order?
![Page 175: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/175.jpg)
Transversal extensions
Let M1,M2 be two matroids on E . The weak order:
M1 ≤w M2 if every independent set in M1 is independent in M2
Fact: the set of all extensions of a matroid M is a lattice under theweak order
Question: is the set of all transversal extensions of a transversalmatroid also a lattice under the weak order?
![Page 176: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/176.jpg)
Transversal extensions
Let M1,M2 be two matroids on E . The weak order:
M1 ≤w M2 if every independent set in M1 is independent in M2
Fact: the set of all extensions of a matroid M is a lattice under theweak order
Question: is the set of all transversal extensions of a transversalmatroid also a lattice under the weak order?
![Page 177: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/177.jpg)
Transversal extensions
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![Page 178: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/178.jpg)
Transversal extensions
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M5 and M6 are transversal extensions of MBut the ordinary join M5 ∨M6 is not transversal
![Page 179: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/179.jpg)
Transversal extensions
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M5 and M6 are transversal extensions of MBut the ordinary join M5 ∨M6 is not transversal
![Page 180: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/180.jpg)
The cyclic ordering problem
Question (Gabow 76)Let B1,B2 be bases of a matroid MIs there an ordering of the elements
b1, b2, . . . , br , br+1, . . . , b2r
such that
- B1 = {b1, . . . , br},- B2 = {br+1, . . . , b2r}, and
- every r cyclically consecutive elements are a basis of M?
It does hold for
I matroids of rank ≤ 4 (Kotlar and Ziv 13)
I graphic matroids (Cordovil and Moreira 93)
I sparse paving matroids (Bonin 13)
I transvesal matroids
![Page 181: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/181.jpg)
The cyclic ordering problem
Question (Gabow 76)Let B1,B2 be bases of a matroid MIs there an ordering of the elements
b1, b2, . . . , br , br+1, . . . , b2r
such that
- B1 = {b1, . . . , br},- B2 = {br+1, . . . , b2r}, and
- every r cyclically consecutive elements are a basis of M?
It does hold for
I matroids of rank ≤ 4 (Kotlar and Ziv 13)
I graphic matroids (Cordovil and Moreira 93)
I sparse paving matroids (Bonin 13)
I transvesal matroids
![Page 182: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/182.jpg)
The cyclic ordering problem
Question (Gabow 76)Let B1,B2 be bases of a matroid MIs there an ordering of the elements
b1, b2, . . . , br , br+1, . . . , b2r
such that
- B1 = {b1, . . . , br},- B2 = {br+1, . . . , b2r}, and
- every r cyclically consecutive elements are a basis of M?
It does hold for
I matroids of rank ≤ 4 (Kotlar and Ziv 13)
I graphic matroids (Cordovil and Moreira 93)
I sparse paving matroids (Bonin 13)
I transvesal matroids
![Page 183: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/183.jpg)
The cyclic ordering problem
It is true for transvesal matroids:
Let M = M[(A1, . . . ,Ar )] and B1,B2 bases of M
Suppose B1 = {b1, . . . , br} with bi ∈ Ai for all iand B2 = {c1, . . . , cr} with ci ∈ Ai for all i
Then every r cyclically consecutive elements of
b1, . . . , br , c1, . . . , cr
form a transversal of (A1, . . . ,Ar ) and thus a basis
(In general, it is true for the larger class of strongly-base oderablematroidsFor more basis-exchange problems, see Bonin 13)
![Page 184: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/184.jpg)
The cyclic ordering problem
It is true for transvesal matroids:
Let M = M[(A1, . . . ,Ar )] and B1,B2 bases of M
Suppose B1 = {b1, . . . , br} with bi ∈ Ai for all iand B2 = {c1, . . . , cr} with ci ∈ Ai for all i
Then every r cyclically consecutive elements of
b1, . . . , br , c1, . . . , cr
form a transversal of (A1, . . . ,Ar ) and thus a basis
(In general, it is true for the larger class of strongly-base oderablematroidsFor more basis-exchange problems, see Bonin 13)
![Page 185: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/185.jpg)
The cyclic ordering problem
It is true for transvesal matroids:
Let M = M[(A1, . . . ,Ar )] and B1,B2 bases of M
Suppose B1 = {b1, . . . , br} with bi ∈ Ai for all iand B2 = {c1, . . . , cr} with ci ∈ Ai for all i
Then every r cyclically consecutive elements of
b1, . . . , br , c1, . . . , cr
form a transversal of (A1, . . . ,Ar ) and thus a basis
(In general, it is true for the larger class of strongly-base oderablematroidsFor more basis-exchange problems, see Bonin 13)
![Page 186: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/186.jpg)
The cyclic ordering problem
It is true for transvesal matroids:
Let M = M[(A1, . . . ,Ar )] and B1,B2 bases of M
Suppose B1 = {b1, . . . , br} with bi ∈ Ai for all iand B2 = {c1, . . . , cr} with ci ∈ Ai for all i
Then every r cyclically consecutive elements of
b1, . . . , br , c1, . . . , cr
form a transversal of (A1, . . . ,Ar ) and thus a basis
(In general, it is true for the larger class of strongly-base oderablematroidsFor more basis-exchange problems, see Bonin 13)
![Page 187: Anna de Mier - UPC Universitat Politècnica de Catalunya · Transversal matroids Anna de Mier Universitat Polit ecnica de Catalunya Barcelona, Spain Let’s Matroid CUSO doctoral](https://reader030.vdocuments.mx/reader030/viewer/2022020303/5afe73e87f8b9a814d8f0ca8/html5/thumbnails/187.jpg)
References from the textEdmonds and Fulkerson 65: Transversals and matroid partition, J. Res.Nat. Bur. Standards Sect. B 69BPiff and Welsh 70: On the vector representations of matroids, J. LondonMath. Soc 2Bonin, de Mier and Noy 03: Lattice path matroids: enumerative aspectsand Tutte polynomials, J. Combin. Theory Ser. A 104Bonin and de Mier 06: Lattice path matroids: structural properties, Eur.J. Combin. 27Simoes-Pereira 72: On subgraphs as matroid cells, Math. Z. 127Matthews 77: Bicircular matroids, Quart. J. Math. 28Brylawski 75: An affine representation for transversal matroids, Studies inAppl. Math. 54Bondy and Welsh 71: Some results on transversal matroids andconstructions for indentically self-dual matroids, Quart. J. Math. 22Bondy 72: Presentations of transversal matroids, J. London Math. Soc 5Mason 69: Representations of Independence Spaces (PhD Dissertation)Bonin 10: Lattice path matroids: the excluded minors J. Combin. TheorySer. B 100Mason 71: A characterization of transversal independence spaces, in:Theorie des Matroıdes (Lecture Notes in Math., 211)
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References from the textIngleton 77: Transversal matroids and related structures, in: HigherCombinatorics (Proc. NATO Adv. Study Inst.)Bonin, Kung and de Mier 11: Characterizations of transversal andfundamental transversal matrids, Elec. J. Combin.Ingleton and Piff 72: Gammoids and Transversal Matroids, J. Combin.Theory Ser. B 15Elizalde and Rubey 13: Bijections for pairs of non-crossing lattice pathsand walks in the plane, preprintCrapo 65: Single-element extensions of matroids, J. Res. Nat. Bur.Standards Sect. B 69Bonin and de Mier 15: Extensions and presentations of transversalmatroids, Eur. J. Combin.Gabow 76: Decomposing symmetric exchanges in matroid bases, Math.Programming 10Kotlar and Ziv 13: On serial symmetric exchanges of matroid bases, J.Graph Theory 73Cordovil and Moreira 93: Bases-cobases grahs and polytopes of matroids,Combinatorica 13Bonin 13: Basis-exchange properties of sparse paving matorids, Adv.Appl. Math. 50
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Some general references about transversal matroids
J. Bonin, An introduction to transversal matroids (lecture notes online),2010R. Brualdi, Transversal matroids, in Combinatorial geometries, CambridgeUniv. Press, 1987J. Oxley, Matroid Theory, 2n ed, Oxford Univ. Press, 2011 [Sections 1.6,2.4, 11.2]