What is a Laminar Matroid?
What is a Laminar Matroid? Tara Fife, James Oxley Department of Mathematics Louisiana State University Baton Rouge Louisiana
Cumberland Conference, May, 2017
Laminar Family
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
2 / 20
Laminar Family
A family A of sets is laminar if for all A1 , A2 ∈ A , either A1 ∩ A2 = ∅ or Ai ⊆ Aj , for distinct i, j ∈ {1, 2}.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
2 / 20
Island Example
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
3 / 20
Independence
We call a set I independent If |I ∩ A| ≤ c(A) for each A ∈ A .
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
4 / 20
Independence We call a set I independent If |I ∩ A| ≤ c(A) for each A ∈ A .
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
4 / 20
Independence We call a set I independent If |I ∩ A| ≤ c(A) for each A ∈ A .
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
4 / 20
Geometric Presentation The following are dependent sets. • Two dots on a point.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
5 / 20
Geometric Presentation The following are dependent sets. • Two dots on a point. • Three dots on a line.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
5 / 20
Geometric Presentation
The following are dependent sets. • Two dots on a point. • Three dots on a line. • Four dots on a plane.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
5 / 20
Geometric Presentation
The following are dependent sets. • Two dots on a point. • Three dots on a line. • Four dots on a plane. • Five dots in space.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
5 / 20
What is a Minor?
Delete e: Remove e.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
6 / 20
What is a Minor?
Delete e: Remove e. Delete 1
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
6 / 20
What is a Minor?
Delete e: Remove e. Delete 1
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
6 / 20
What is a Minor?
Delete e: Remove e. Contract e: Project from e onto a hyperplane that does not contain e.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
6 / 20
What is a Minor?
Delete e: Remove e. Contract e: Project from e onto a hyperplane that does not contain e. Contract 1
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
6 / 20
What is a Minor?
Delete e: Remove e. Contract e: Project from e onto a hyperplane that does not contain e. Contract 1
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
6 / 20
What is a Minor?
Delete e: Remove e. Contract e: Project from e onto a hyperplane that does not contain e. Contract 1
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
6 / 20
What is a Minor?
Delete e: Remove e. Contract e: Project from e onto a hyperplane that does not contain e. Delete 5
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
6 / 20
What is a Minor?
Delete e: Remove e. Contract e: Project from e onto a hyperplane that does not contain e. Delete 5
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
6 / 20
What is a Minor?
Delete e: Remove e. Contract e: Project from e onto a hyperplane that does not contain e. Contract 5
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
6 / 20
What is a Minor?
Delete e: Remove e. Contract e: Project from e onto a hyperplane that does not contain e. Contract 5
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
6 / 20
What is a Minor?
Delete e: Remove e. Contract e: Project from e onto a hyperplane that does not contain e. Contract 5
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
6 / 20
Matroids
A matroid is a nice notion of independence and dependence in a finite set.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
7 / 20
Geometric Representation of a Laminar Matroid
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
8 / 20
Geometric Representation of a Laminar Matroid
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
8 / 20
Geometric Representation of a Laminar Matroid
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
8 / 20
Geometric Representation of a Laminar Matroid
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
8 / 20
Geometric Representation of a Laminar Matroid
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
8 / 20
Minors of Laminar Matroids
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
9 / 20
Minors of Laminar Matroids
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
9 / 20
Not Laminar
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
10 / 20
An Excluded Minor
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
11 / 20
The Excluded Minors
Theorem The excluded minors of laminar matroids are:
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
12 / 20
Nested Matroids These are laminar matroids which have a representation where the family A looks like a path.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
13 / 20
Nested Matroids These are laminar matroids which have a representation where the family A looks like a path.
Theorem (O., Prendergast, and Row) The excluded minors of nested matroids are:
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
13 / 20
Another Look at Circuits
A circuit is a minimal dependent set.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
14 / 20
Another Look at Circuits
A circuit is a minimal dependent set. {1, 2, 3}
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
14 / 20
Another Look at Circuits
A circuit is a minimal dependent set. {1, 2, 3}, {1, 2, 4, 5}, {1, 3, 4, 5}, {2, 3, 4, 5}
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
14 / 20
Another Look at Circuits
A circuit is a minimal dependent set. {1, 2, 3}, {1, 2, 4, 5}, {1, 3, 4, 5}, {2, 3, 4, 5}, {1, 2, 4, 6, 7}, {1, 2, 5, 6, 7}, {1, 3, 4, 6, 7}, {1, 3, 5, 6, 7}, {2, 3, 4, 6, 7}, {2, 3, 5, 6, 7}
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
14 / 20
Another Look at Circuits
A circuit is a minimal dependent set. {1, 2, 3}, {1, 2, 4, 5}, {1, 3, 4, 5}, {2, 3, 4, 5}, {1, 2, 4, 6, 7}, {1, 2, 5, 6, 7}, {1, 3, 4, 6, 7}, {1, 3, 5, 6, 7}, {2, 3, 4, 6, 7}, {2, 3, 5, 6, 7}, etc.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
14 / 20
Another Look at Circuits
A Circuit is a minimally dependent set.
Theorem For a laminar matroid M(E , A , c), a set C is a circuit if it is a minimal set such that C ⊆ A and |C | = c(A) + 1 for some A ∈ A .
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
15 / 20
Hamiltonian Flats
If X ⊆ E , we define cl(X ), the closure of X , to be X ∪ {e : there is a circuit C with e ∈ C ⊆ e ∪ X }.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
16 / 20
Hamiltonian Flats
If X ⊆ E , we define cl(X ), the closure of X , to be X ∪ {e : there is a circuit C with e ∈ C ⊆ e ∪ X }.
A Hamiltonian flat is the closure of a circuit.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
16 / 20
Hamiltonian Flats
If X ⊆ E , we define cl(X ), the closure of X , to be X ∪ {e : there is a circuit C with e ∈ C ⊆ e ∪ X }.
A Hamiltonian flat is the closure of a circuit. Our Hamiltonian flats are:
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
16 / 20
Hamiltonian Flats
If X ⊆ E , we define cl(X ), the closure of X , to be X ∪ {e : there is a circuit C with e ∈ C ⊆ e ∪ X }.
A Hamiltonian flat is the closure of a circuit. Our Hamiltonian flats are: {1, 2, 3}, {1, 2, 3, 4, 5}, {1, 2, 3, 4, 5, 6, 7}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
16 / 20
Hamiltonian Flats
A Hamiltonian flat is the closure of a circuit.
Our Hamiltonian flats are:
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
17 / 20
Hamiltonian Flats
A Hamiltonian flat is the closure of a circuit.
Our Hamiltonian flats are: {1, 2, 3}, {1, 2, 3, 4, 5, 6}, {1, 2, 3, 7, 8}, {1, 2, 3, 7, 8, 9, 10}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
17 / 20
Hamiltonian Flats
Theorem A matroid is nested if and only if its Hamiltonian flats form a chain under inclusion.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
18 / 20
Hamiltonian Flats
Theorem A matroid is nested if and only if its Hamiltonian flats form a chain under inclusion.
Theorem A matroid M is laminar if and only if, for every independent set X of size 1, the Hamiltonian flats of M containing X form a chain under inclusion.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
18 / 20
A Generalization
Let Mk be the class of all matroids such that for every independent set X of size k, the Hamiltonian flats of M containing X form a chain under inclusion.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
19 / 20
A Generalization
Let Mk be the class of all matroids such that for every independent set X of size k, the Hamiltonian flats of M containing X form a chain under inclusion.
• M0 is the class of nested matroids.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
19 / 20
A Generalization
Let Mk be the class of all matroids such that for every independent set X of size k, the Hamiltonian flats of M containing X form a chain under inclusion.
• M0 is the class of nested matroids. • M1 is the class of laminar matroids.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
19 / 20
A Generalization
Let Mk be the class of all matroids such that for every independent set X of size k, the Hamiltonian flats of M containing X form a chain under inclusion.
• M0 is the class of nested matroids. • M1 is the class of laminar matroids. • M2 is minor-closed, and its excluded minors are known.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
19 / 20
A Generalization
Let Mk be the class of all matroids such that for every independent set X of size k, the Hamiltonian flats of M containing X form a chain under inclusion.
• M0 is the class of nested matroids. • M1 is the class of laminar matroids. • M2 is minor-closed, and its excluded minors are known. • M3 is minor-closed.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
19 / 20
A Generalization
Let Mk be the class of all matroids such that for every independent set X of size k, the Hamiltonian flats of M containing X form a chain under inclusion.
• M0 is the class of nested matroids. • M1 is the class of laminar matroids. • M2 is minor-closed, and its excluded minors are known. • M3 is minor-closed. • Mk is not minor-closed for any k ≥ 4.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
19 / 20
Thank You
Thank You.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
20 / 20
Cumberland Conference, May, 2017
Laminar Family
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
2 / 20
Laminar Family
A family A of sets is laminar if for all A1 , A2 ∈ A , either A1 ∩ A2 = ∅ or Ai ⊆ Aj , for distinct i, j ∈ {1, 2}.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
2 / 20
Island Example
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
3 / 20
Independence
We call a set I independent If |I ∩ A| ≤ c(A) for each A ∈ A .
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
4 / 20
Independence We call a set I independent If |I ∩ A| ≤ c(A) for each A ∈ A .
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
4 / 20
Independence We call a set I independent If |I ∩ A| ≤ c(A) for each A ∈ A .
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
4 / 20
Geometric Presentation The following are dependent sets. • Two dots on a point.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
5 / 20
Geometric Presentation The following are dependent sets. • Two dots on a point. • Three dots on a line.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
5 / 20
Geometric Presentation
The following are dependent sets. • Two dots on a point. • Three dots on a line. • Four dots on a plane.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
5 / 20
Geometric Presentation
The following are dependent sets. • Two dots on a point. • Three dots on a line. • Four dots on a plane. • Five dots in space.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
5 / 20
What is a Minor?
Delete e: Remove e.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
6 / 20
What is a Minor?
Delete e: Remove e. Delete 1
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
6 / 20
What is a Minor?
Delete e: Remove e. Delete 1
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
6 / 20
What is a Minor?
Delete e: Remove e. Contract e: Project from e onto a hyperplane that does not contain e.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
6 / 20
What is a Minor?
Delete e: Remove e. Contract e: Project from e onto a hyperplane that does not contain e. Contract 1
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
6 / 20
What is a Minor?
Delete e: Remove e. Contract e: Project from e onto a hyperplane that does not contain e. Contract 1
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
6 / 20
What is a Minor?
Delete e: Remove e. Contract e: Project from e onto a hyperplane that does not contain e. Contract 1
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
6 / 20
What is a Minor?
Delete e: Remove e. Contract e: Project from e onto a hyperplane that does not contain e. Delete 5
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
6 / 20
What is a Minor?
Delete e: Remove e. Contract e: Project from e onto a hyperplane that does not contain e. Delete 5
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
6 / 20
What is a Minor?
Delete e: Remove e. Contract e: Project from e onto a hyperplane that does not contain e. Contract 5
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
6 / 20
What is a Minor?
Delete e: Remove e. Contract e: Project from e onto a hyperplane that does not contain e. Contract 5
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
6 / 20
What is a Minor?
Delete e: Remove e. Contract e: Project from e onto a hyperplane that does not contain e. Contract 5
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
6 / 20
Matroids
A matroid is a nice notion of independence and dependence in a finite set.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
7 / 20
Geometric Representation of a Laminar Matroid
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
8 / 20
Geometric Representation of a Laminar Matroid
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
8 / 20
Geometric Representation of a Laminar Matroid
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
8 / 20
Geometric Representation of a Laminar Matroid
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
8 / 20
Geometric Representation of a Laminar Matroid
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
8 / 20
Minors of Laminar Matroids
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
9 / 20
Minors of Laminar Matroids
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
9 / 20
Not Laminar
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
10 / 20
An Excluded Minor
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
11 / 20
The Excluded Minors
Theorem The excluded minors of laminar matroids are:
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
12 / 20
Nested Matroids These are laminar matroids which have a representation where the family A looks like a path.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
13 / 20
Nested Matroids These are laminar matroids which have a representation where the family A looks like a path.
Theorem (O., Prendergast, and Row) The excluded minors of nested matroids are:
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
13 / 20
Another Look at Circuits
A circuit is a minimal dependent set.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
14 / 20
Another Look at Circuits
A circuit is a minimal dependent set. {1, 2, 3}
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
14 / 20
Another Look at Circuits
A circuit is a minimal dependent set. {1, 2, 3}, {1, 2, 4, 5}, {1, 3, 4, 5}, {2, 3, 4, 5}
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
14 / 20
Another Look at Circuits
A circuit is a minimal dependent set. {1, 2, 3}, {1, 2, 4, 5}, {1, 3, 4, 5}, {2, 3, 4, 5}, {1, 2, 4, 6, 7}, {1, 2, 5, 6, 7}, {1, 3, 4, 6, 7}, {1, 3, 5, 6, 7}, {2, 3, 4, 6, 7}, {2, 3, 5, 6, 7}
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
14 / 20
Another Look at Circuits
A circuit is a minimal dependent set. {1, 2, 3}, {1, 2, 4, 5}, {1, 3, 4, 5}, {2, 3, 4, 5}, {1, 2, 4, 6, 7}, {1, 2, 5, 6, 7}, {1, 3, 4, 6, 7}, {1, 3, 5, 6, 7}, {2, 3, 4, 6, 7}, {2, 3, 5, 6, 7}, etc.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
14 / 20
Another Look at Circuits
A Circuit is a minimally dependent set.
Theorem For a laminar matroid M(E , A , c), a set C is a circuit if it is a minimal set such that C ⊆ A and |C | = c(A) + 1 for some A ∈ A .
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
15 / 20
Hamiltonian Flats
If X ⊆ E , we define cl(X ), the closure of X , to be X ∪ {e : there is a circuit C with e ∈ C ⊆ e ∪ X }.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
16 / 20
Hamiltonian Flats
If X ⊆ E , we define cl(X ), the closure of X , to be X ∪ {e : there is a circuit C with e ∈ C ⊆ e ∪ X }.
A Hamiltonian flat is the closure of a circuit.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
16 / 20
Hamiltonian Flats
If X ⊆ E , we define cl(X ), the closure of X , to be X ∪ {e : there is a circuit C with e ∈ C ⊆ e ∪ X }.
A Hamiltonian flat is the closure of a circuit. Our Hamiltonian flats are:
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
16 / 20
Hamiltonian Flats
If X ⊆ E , we define cl(X ), the closure of X , to be X ∪ {e : there is a circuit C with e ∈ C ⊆ e ∪ X }.
A Hamiltonian flat is the closure of a circuit. Our Hamiltonian flats are: {1, 2, 3}, {1, 2, 3, 4, 5}, {1, 2, 3, 4, 5, 6, 7}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
16 / 20
Hamiltonian Flats
A Hamiltonian flat is the closure of a circuit.
Our Hamiltonian flats are:
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
17 / 20
Hamiltonian Flats
A Hamiltonian flat is the closure of a circuit.
Our Hamiltonian flats are: {1, 2, 3}, {1, 2, 3, 4, 5, 6}, {1, 2, 3, 7, 8}, {1, 2, 3, 7, 8, 9, 10}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
17 / 20
Hamiltonian Flats
Theorem A matroid is nested if and only if its Hamiltonian flats form a chain under inclusion.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
18 / 20
Hamiltonian Flats
Theorem A matroid is nested if and only if its Hamiltonian flats form a chain under inclusion.
Theorem A matroid M is laminar if and only if, for every independent set X of size 1, the Hamiltonian flats of M containing X form a chain under inclusion.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
18 / 20
A Generalization
Let Mk be the class of all matroids such that for every independent set X of size k, the Hamiltonian flats of M containing X form a chain under inclusion.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
19 / 20
A Generalization
Let Mk be the class of all matroids such that for every independent set X of size k, the Hamiltonian flats of M containing X form a chain under inclusion.
• M0 is the class of nested matroids.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
19 / 20
A Generalization
Let Mk be the class of all matroids such that for every independent set X of size k, the Hamiltonian flats of M containing X form a chain under inclusion.
• M0 is the class of nested matroids. • M1 is the class of laminar matroids.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
19 / 20
A Generalization
Let Mk be the class of all matroids such that for every independent set X of size k, the Hamiltonian flats of M containing X form a chain under inclusion.
• M0 is the class of nested matroids. • M1 is the class of laminar matroids. • M2 is minor-closed, and its excluded minors are known.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
19 / 20
A Generalization
Let Mk be the class of all matroids such that for every independent set X of size k, the Hamiltonian flats of M containing X form a chain under inclusion.
• M0 is the class of nested matroids. • M1 is the class of laminar matroids. • M2 is minor-closed, and its excluded minors are known. • M3 is minor-closed.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
19 / 20
A Generalization
Let Mk be the class of all matroids such that for every independent set X of size k, the Hamiltonian flats of M containing X form a chain under inclusion.
• M0 is the class of nested matroids. • M1 is the class of laminar matroids. • M2 is minor-closed, and its excluded minors are known. • M3 is minor-closed. • Mk is not minor-closed for any k ≥ 4.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
19 / 20
Thank You
Thank You.
Tara Fife, James Oxley (LSU)
What is a Laminar Matroid?
CC2017
20 / 20
