Minimally k -Connected Graphs and Matroids Xiangqian Zhou (Joe) - - PowerPoint PPT Presentation

Minimally k-Connected Graphs and Matroids Xiangqian Zhou (Joe)

Wright State University and Huaqiao University

Minimally k-Connected Graphs and Matroids

Definition of a Matroid

Consider a matrix over GF(2)

   1 2 3 4 5 6 7 1 1 1 1 A = 1 1 1 1 1 1 1 1   

Minimally k-Connected Graphs and Matroids

Definition of a Matroid

Consider a matrix over GF(2)

   1 2 3 4 5 6 7 1 1 1 1 A = 1 1 1 1 1 1 1 1    Let E = {1, 2, 3, 4, 5, 6, 7} and I = {I ⊆ E | Columns in I are independent}.

Minimally k-Connected Graphs and Matroids

Definition of a Matroid

Consider a matrix over GF(2)

   1 2 3 4 5 6 7 1 1 1 1 A = 1 1 1 1 1 1 1 1    Let E = {1, 2, 3, 4, 5, 6, 7} and I = {I ⊆ E | Columns in I are independent}.

• ex. {1, 2, 3} ∈ I and {4, 5, 6} /

∈ I

Minimally k-Connected Graphs and Matroids

Definition of a Matroid, Whitney 1935 (I0) ∅ ∈ I. (I1) If J ⊂ I and I ∈ I, then J ∈ I. (I2) If I, J ∈ I with |I| < |J|, then there exists x ∈ J\I such that I ∪ {x} ∈ I.

Minimally k-Connected Graphs and Matroids

Definition of a Matroid, Whitney 1935 (I0) ∅ ∈ I. (I1) If J ⊂ I and I ∈ I, then J ∈ I. (I2) If I, J ∈ I with |I| < |J|, then there exists x ∈ J\I such that I ∪ {x} ∈ I. A matroid is a pair (E, I) where E is a finite set and I ⊆ P(E) satisfies (I0)-(I2).

Minimally k-Connected Graphs and Matroids

Definition of a Matroid, Whitney 1935 (I0) ∅ ∈ I. (I1) If J ⊂ I and I ∈ I, then J ∈ I. (I2) If I, J ∈ I with |I| < |J|, then there exists x ∈ J\I such that I ∪ {x} ∈ I. A matroid is a pair (E, I) where E is a finite set and I ⊆ P(E) satisfies (I0)-(I2). E is called the ground set. Members of I are called independent sets.

Minimally k-Connected Graphs and Matroids

Examples of Matroids

Minimally k-Connected Graphs and Matroids

Examples of Matroids

F-representable Matroids

A matroid M is F-representable if M is obtained from a matrix over the field F.

Minimally k-Connected Graphs and Matroids

Examples of Matroids

F-representable Matroids

A matroid M is F-representable if M is obtained from a matrix over the field F.

Graphic Matroids

Let G = (V , E) be a graph. Define the cycle matroid of G, denoted by M(G), as follows: Ground set: E(G). Independent sets: Subsets of E(G) that do not contain any cycle of G.

Minimally k-Connected Graphs and Matroids

Minimally k-Connected Graphs A graph G is minimally k-connected if G is k-connected and for every e ∈ E(G), G\e is not k-connected.

Minimally k-Connected Graphs and Matroids

Minimally k-Connected Graphs A graph G is minimally k-connected if G is k-connected and for every e ∈ E(G), G\e is not k-connected.

Halin, 1969

A minimally k-connected graph has a vertex of degree k.

Minimally k-Connected Graphs and Matroids

Minimally k-Connected Graphs A graph G is minimally k-connected if G is k-connected and for every e ∈ E(G), G\e is not k-connected.

Halin, 1969

A minimally k-connected graph has a vertex of degree k.

Mader, 1979

In every minimally k-connected graph G, the number of degree-k vertices is at least (k − 1)|V (G)| + 2k 2k − 1

Minimally k-Connected Graphs and Matroids

From Graphs to Matroids A matroid M is minimally k-connected if M is k-connected and for every e ∈ E(M), M\e is not k-connected.

Minimally k-Connected Graphs and Matroids

From Graphs to Matroids A matroid M is minimally k-connected if M is k-connected and for every e ∈ E(M), M\e is not k-connected.

Degree-k vertices – Cocircuit of size k

If G is a 2-connected loopless graph with at least three vertices, then the set of edges meeting a vertex is a cocircuit in M(G).

Minimally k-Connected Graphs and Matroids

A vertex bond picture

Figure : A vertex bond is a cocircuit

Minimally k-Connected Graphs and Matroids

Minimally k-Connected Matroids

Problem 14.4.9, Matroid Theory by Oxley

Let k ≥ 2. If M is a minimally k-connected matroid with |E(M)| ≥ 2(k − 1), does M have a cocircuit of size k?

Minimally k-Connected Graphs and Matroids

Minimally k-Connected Matroids

Problem 14.4.9, Matroid Theory by Oxley

Let k ≥ 2. If M is a minimally k-connected matroid with |E(M)| ≥ 2(k − 1), does M have a cocircuit of size k?

Murty, 1974

Yes if k = 2.

Minimally k-Connected Graphs and Matroids

Minimally k-Connected Matroids

Problem 14.4.9, Matroid Theory by Oxley

Let k ≥ 2. If M is a minimally k-connected matroid with |E(M)| ≥ 2(k − 1), does M have a cocircuit of size k?

Murty, 1974

Yes if k = 2.

Wong, 1978

Yes if k = 3.

Minimally k-Connected Graphs and Matroids

The Lower Bounds for k = 2, 3

Oxley, 1981

Let M be a minimally 2-connected matroid. Then the number of pairwise disjoint 2-cocircuits is at least

1 3(r(M) + 2)

if |E(M)| < 1

3(4r(M) − 1)

r ∗(M) + 1 if |E(M)| ≥ 1

3(4r(M) − 1)

Minimally k-Connected Graphs and Matroids

The Lower Bounds for k = 2, 3

Oxley, 1981

Let M be a minimally 2-connected matroid. Then the number of pairwise disjoint 2-cocircuits is at least

1 3(r(M) + 2)

if |E(M)| < 1

3(4r(M) − 1)

r ∗(M) + 1 if |E(M)| ≥ 1

3(4r(M) − 1)

Oxley, 1984

A minimally 3-connected matroid M has at least

1 2r ∗(M) + 1 triads (3-cocircuit).

Minimally k-Connected Graphs and Matroids

The case k ≥ 4.

Reid, Wu, and Zhou

Let M be a minimally 4-connected matroid with |E(M)| ≥ 6. Then M has a cocircuit of size 4; or M is isomorphic to a special matroid with nine elements.

Minimally k-Connected Graphs and Matroids

The case k ≥ 4.

Reid, Wu, and Zhou

Let M be a minimally 4-connected matroid with |E(M)| ≥ 6. Then M has a cocircuit of size 4; or M is isomorphic to a special matroid with nine elements. There exists a minimally k-connected matroid with 2k + 1 elements that has no cocircuit of size k.

Minimally k-Connected Graphs and Matroids

k-Separation of a Matroid

k-separating sets

A set A ⊆ E(M) is k-separating if rM(A) + rM(E\A) − r(M) ≤ k − 1

Minimally k-Connected Graphs and Matroids

k-Separation of a Matroid

k-separating sets

A set A ⊆ E(M) is k-separating if rM(A) + rM(E\A) − r(M) ≤ k − 1

k-separations

A partition (A, B) of E(M) is a k-separation if A is k-separating; and |A|, |B| ≥ k.

Minimally k-Connected Graphs and Matroids

k-Separation of a Matroid

k-separating sets

A set A ⊆ E(M) is k-separating if rM(A) + rM(E\A) − r(M) ≤ k − 1

k-separations

A partition (A, B) of E(M) is a k-separation if A is k-separating; and |A|, |B| ≥ k.

n-connected matroids

M is n-connected if M has no k-separation for k < n.

Minimally k-Connected Graphs and Matroids

The Uncrossing Technique If X and Y are both k-separating in M and X ∩ Y is not (k − 1)-separating in M, then X ∪ Y is k-separating in M.

Minimally k-Connected Graphs and Matroids

The Uncrossing Technique If X and Y are both k-separating in M and X ∩ Y is not (k − 1)-separating in M, then X ∪ Y is k-separating in M.

• E\X

X Y E\Y

Figure : The Uncrossing Lemma

Minimally k-Connected Graphs and Matroids

Sketch of the Proof e Ae Be

Minimally k-Connected Graphs and Matroids

Sketch of the Proof e Ae Be f Af Bf

Minimally k-Connected Graphs and Matroids

Sketch of the Proof e Ae Be f Af Bf {g} size 2 size 2

Figure : Crossing 4-separations

Minimally k-Connected Graphs and Matroids

Restricting the size

Minimally k-Connected Graphs and Matroids

Restricting the size Let M be a minimally 4-connected matroid that has no cocircuit of size 4. Then |E(M)| = 9; or M has a tripod.

Minimally k-Connected Graphs and Matroids

Restricting the size Let M be a minimally 4-connected matroid that has no cocircuit of size 4. Then |E(M)| = 9; or M has a tripod.

✘✘✘✘✘✘

• ◗

◗ ◗ ◗ ◗ s s s s s s s s s

e f g e1 e2 f1 f2 g1 g2

Minimally k-Connected Graphs and Matroids

Restricting the size Let M be a minimally 4-connected matroid that has no cocircuit of size 4. Then |E(M)| = 9; or M has a tripod.

✘✘✘✘✘✘

• ◗

◗ ◗ ◗ ◗ s s s s s s s s s

e f g e1 e2 f1 f2 g1 g2 If M has a tripod, then |E(M)| = 9.

Minimally k-Connected Graphs and Matroids

Minimally k-connected matroids with 2k + 1 elements

k-splitting family

Let |E| = 2k + 1. F ⊆ Pk(E) is a k-splitting family if s1) |F ∩ F ′| ≤ k − 2 for distinct F, F ′ ∈ F; and s2) For each e ∈ E, there exist Ae, Be ∈ F, such that (Ae, Be) partition E\{e}.

Minimally k-Connected Graphs and Matroids

Minimally k-connected matroids with 2k + 1 elements

k-splitting family

Let |E| = 2k + 1. F ⊆ Pk(E) is a k-splitting family if s1) |F ∩ F ′| ≤ k − 2 for distinct F, F ′ ∈ F; and s2) For each e ∈ E, there exist Ae, Be ∈ F, such that (Ae, Be) partition E\{e}.

The following are equivalent:

There exists a minimally k-connected matroid with 2k + 1 elements with no k-element cocircuits. There exists a k-splitting family.

Minimally k-Connected Graphs and Matroids

4-splitting family |E| = 9 and F ⊆ P4(E) such that |F ∩ F ′| ≤ 2 for F = F ′ ∈ F; and For every e, there exist Ae, Be ∈ F with (Ae, Be) partitioning E\{e}

Minimally k-Connected Graphs and Matroids

4-splitting family |E| = 9 and F ⊆ P4(E) such that |F ∩ F ′| ≤ 2 for F = F ′ ∈ F; and For every e, there exist Ae, Be ∈ F with (Ae, Be) partitioning E\{e}

Reid, Wu, and Zhou

Every 4-splitting family is a 2-(9, 4, 3)-design.

Minimally k-Connected Graphs and Matroids

The unique 4-splitting family. x Ax Bx 1 {2, 3, 4, 5} {6, 7, 8, 9} 2 {1, 3, 6, 7} {4, 5, 8, 9} 3 {1, 2, 8, 9} {4, 5, 6, 7} 4 {1, 5, 6, 8} {2, 3, 7, 9} 5 {1, 4, 7, 9} {2, 3, 6, 8} 6 {2, 4, 7, 8} {1, 3, 5, 9} 7 {2, 5, 6, 9} {1, 3, 4, 8} 8 {3, 4, 6, 9} {1, 2, 5, 7} 9 {3, 5, 7, 8} {1, 2, 4, 6}

Minimally k-Connected Graphs and Matroids

The cases k ≥ 5

Reid,Wu, and Zhou

For k ≥ 5, there exists a minimally k-connected matroid with 2k + 1 elements that has no cocircuit of size k.

Minimally k-Connected Graphs and Matroids

The cases k ≥ 5

Reid,Wu, and Zhou

For k ≥ 5, there exists a minimally k-connected matroid with 2k + 1 elements that has no cocircuit of size k.

Proof by Induction on k.

x A′

x = Ax + e or f

B′

x = Bx + f or e

e A′

e = Ae + f

B′

e

f A′

f = Af + e

B′

f

Minimally k-Connected Graphs and Matroids

The uniform Ray-Chaudhuri-Wilson Inequality

Let |E| = n and H ⊆ Pk(E).

Let L ⊆ Z≥0 be a finite set with |L| = s. If for H1 = H2 ∈ H, |H1 ∩ H2| ∈ L, then |H| ≤ n

s

• .

The equality holds if and only if H is a 2s-design.

Minimally k-Connected Graphs and Matroids

A New Conjecture

Minimally k-Connected Graphs and Matroids

A New Conjecture For k ≥ 5, if M is a minimally k-connected matroid with |E(M)| ≥ 2(k − 1) and |E(M)| = 2k + 1, then M has a cocircuit a size k.

Minimally k-Connected Graphs and Matroids

Recent Progress

Minimally k-Connected Graphs and Matroids

Recent Progress

A Corollary

A minimally 4-connected matroid has a cocircuit of size 4

• r 5.

Minimally k-Connected Graphs and Matroids

Recent Progress

A Corollary

A minimally 4-connected matroid has a cocircuit of size 4

• r 5.

Costalonga, Deng, and Zhou 2017

For k ≥ 4, a minimally k-connected matroid has a cocircuit of size at most 2k − 3.

Minimally k-Connected Graphs and Matroids

Critical Circuits

Minimally k-Connected Graphs and Matroids

Critical Circuits C is a critical circuit of a k-connected matroid M if M\e is not k-connected for every e ∈ C.

Minimally k-Connected Graphs and Matroids

Critical Circuits C is a critical circuit of a k-connected matroid M if M\e is not k-connected for every e ∈ C.

Tutte’s Triangle Lemma

A critical triangle meets at least two triads.

Minimally k-Connected Graphs and Matroids

Critical Circuits C is a critical circuit of a k-connected matroid M if M\e is not k-connected for every e ∈ C.

Tutte’s Triangle Lemma

A critical triangle meets at least two triads.

Lemos, 1989

A critical circuit in a 3-connected matroid meets at least two triads.

Minimally k-Connected Graphs and Matroids

Critical Circuits C is a critical circuit of a k-connected matroid M if M\e is not k-connected for every e ∈ C.

Tutte’s Triangle Lemma

A critical triangle meets at least two triads.

Lemos, 1989

A critical circuit in a 3-connected matroid meets at least two triads.

Costalonga, Deng, and Zhou 2017

For k ≥ 4, a critical circuit in a k-connected matroid meets a cocircuit of size at most 2k − 3.

Minimally k-Connected Graphs and Matroids