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 0 0 1 0 1 1 A = 0 1 0 1 1 0 1     0 0 1 0 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 0 0 1 0 1 1 A = 0 1 0 1 1 0 1     0 0 1 0 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 0 0 1 0 1 1 A = 0 1 0 1 1 0 1     0 0 1 0 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 ) | + 2 k 2 k − 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 | E ( M ) | < 1 3 (4 r ( M ) − 1) 3 ( r ( M ) + 2) if r ∗ ( M ) + 1 | E ( M ) | ≥ 1 if 3 (4 r ( 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 | E ( M ) | < 1 3 (4 r ( M ) − 1) 3 ( r ( M ) + 2) if r ∗ ( M ) + 1 | E ( M ) | ≥ 1 if 3 (4 r ( M ) − 1) Oxley, 1984 A minimally 3-connected matroid M has at least 2 r ∗ ( M ) + 1 triads (3-cocircuit). 1 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 2 k + 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 r M ( A ) + r M ( 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 r M ( A ) + r M ( 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 r M ( A ) + r M ( 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 . X E\X ������ ������ ������ ������ ������ ������ ������ ������ Y ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ E\Y ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ Figure : The Uncrossing Lemma Minimally k -Connected Graphs and Matroids

Sketch of the Proof A e B e e Minimally k -Connected Graphs and Matroids

Sketch of the Proof A e B e e A f f B f Minimally k -Connected Graphs and Matroids

Sketch of the Proof A e B e e { g } A f size 2 f B f 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. ◗ ✘✘✘✘✘✘ f ◗ e ◗ s f 1 ◗ � ◗ s s s g 1 � g f 2 s � s g 2 s e 1 s e 2 s Minimally k -Connected Graphs and Matroids