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Representations of even cut matroids Irene Pivotto Department of - - PowerPoint PPT Presentation

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Representations of even cut matroids Irene Pivotto Department of Combinatorics and Optimization University of Waterloo January 8, 2010 Joint work with B. Guenin Irene Pivotto (UW) January 8, 2010 1 / 31 Outline 1 Motivation 2 Definitions 3

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Representations of even cut matroids

Irene Pivotto

Department of Combinatorics and Optimization University of Waterloo

January 8, 2010

Joint work with B. Guenin

Irene Pivotto (UW) January 8, 2010 1 / 31

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SLIDE 2

Outline

1 Motivation 2 Definitions 3 What we want to do (and did) 4 How we want to use it 5 Some work left to do Irene Pivotto (UW) January 8, 2010 2 / 31

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Motivation

Why even cut matroids?

  • minor closed class
  • contains cographic matroids (cut matroids)
  • may help proving Seymour’s conjecture on 1-flowing matroids: when

does the max flow-min cut theorem extends to binary matroids?

Irene Pivotto (UW) January 8, 2010 3 / 31

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Definitions

G: graph with labeled edges. The cut-matroid represented by G has: elements - edges of G cycles - cuts of G Denoted by cut(G, T).

Irene Pivotto (UW) January 8, 2010 4 / 31

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Definitions

G: graph with labeled edges. The cut-matroid represented by G has: elements - edges of G cycles - cuts of G Denoted by cut(G, T). Example

1 8 4 7 3 2 6 9 5

The elements of cut(G) are {1, 2, . . . , 9} 1, 2, 5, 6 is a cycle of cut(G)

Irene Pivotto (UW) January 8, 2010 4 / 31

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Definitions

For cut matroids we know: (1) excluded minors (Tutte 1959) (2) “unique” representation: Whitney-flips (Whitney 1933) G ∼W G ′

G G'

Irene Pivotto (UW) January 8, 2010 5 / 31

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Definitions

Example

1 4 12 2 6 5 7 11 13 9 8 10 3 1 4 12 2 6 5 7 11 13 9 8 10 3

Irene Pivotto (UW) January 8, 2010 6 / 31

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Definitions

Whitney-flips preserve cuts:

1 4 12 2 6 5 7 11 13 9 8 10 3 1 4 12 2 6 5 7 11 13 9 8 10 3

Irene Pivotto (UW) January 8, 2010 7 / 31

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Definitions

Whitney-flips preserve cuts:

1 4 12 2 6 5 7 11 13 9 8 10 3 1 4 12 2 6 5 7 11 13 9 8 10 3

same cuts ⇔ related by ∼W

Irene Pivotto (UW) January 8, 2010 7 / 31

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Definitions

Graft: pair (G, T) where G is a graph and T ⊆ VG, |T| even. A cut δ(U) ⊆ EG is even (resp. odd) if |U ∩ T| is even (resp. odd). The even cut matroid represented by (G, T) has elements - edges of G cycles - even cuts of (G, T) Denoted by ecut(G, T)

Irene Pivotto (UW) January 8, 2010 8 / 31

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Definitions

Graft: pair (G, T) where G is a graph and T ⊆ VG, |T| even. A cut δ(U) ⊆ EG is even (resp. odd) if |U ∩ T| is even (resp. odd). The even cut matroid represented by (G, T) has elements - edges of G cycles - even cuts of (G, T) Denoted by ecut(G, T) Note: cut matroids are even cut matroids.

Irene Pivotto (UW) January 8, 2010 8 / 31

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Definitions

Example

1 8 4 7 3 2 6 9 5

Boxed vertices are in T

Irene Pivotto (UW) January 8, 2010 9 / 31

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Definitions

Example

1 8 4 7 3 2 6 9 5

Boxed vertices are in T 1, 4, 7 is an odd cut

Irene Pivotto (UW) January 8, 2010 10 / 31

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Definitions

Example

1 8 4 7 3 2 6 9 5

Boxed vertices are in T 1, 4, 7 is an odd cut 1, 2, 5, 6 is an even cut

Irene Pivotto (UW) January 8, 2010 11 / 31

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Definitions

Example

1 8 4 7 3 2 6 9 5

Boxed vertices are in T 1, 4, 7 is an odd cut 1, 2, 5, 6 is an even cut 1, 4, 7, 6, 3, 8 is an even cut

Irene Pivotto (UW) January 8, 2010 12 / 31

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Idea for finding excluded minors

M : class of even cut matroids (a) Show: if M ∈ M, then M contains one of N1, . . . , Nk (b) For all i, characterize matroids M minimally not in M with Ni ≤ M.

Irene Pivotto (UW) January 8, 2010 13 / 31

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Idea for finding excluded minors

M : class of even cut matroids (a) Show: if M ∈ M, then M contains one of N1, . . . , Nk (b) For all i, characterize matroids M minimally not in M with Ni ≤ M. (a) We can pick excluded minors for cographic matroids

√

Irene Pivotto (UW) January 8, 2010 13 / 31

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Idea for finding excluded minors

M : class of even cut matroids (a) Show: if M ∈ M, then M contains one of N1, . . . , Nk (b) For all i, characterize matroids M minimally not in M with Ni ≤ M. (a) We can pick excluded minors for cographic matroids

√

(b) We want to use the fact that even cut matroids are represented by grafts. Hard: even cut matroids have many representations.

Irene Pivotto (UW) January 8, 2010 13 / 31

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Many representations

Example

1 8 10 7 9 6 14 13 12 11 15 5 4 3 2 9 2 7 10 1 6 14 13 15 11 12 5 4 3 8

Irene Pivotto (UW) January 8, 2010 14 / 31

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Minor operations

Matroid minors correspond to graft minors: M = ecut(G, T) ⇔ (G, T)

matroid minor

⇓ ⇓

graft minor

N = ecut(H, S) ⇔ (H, S)

Irene Pivotto (UW) January 8, 2010 15 / 31

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Minor operations Matroid contraction = graft deletion

M/e ⇒ (G \ e, T)

1 8 4 7 3 2 6 9 5

⇒

1 8 4 7 3 6 9 5

e = 2

Irene Pivotto (UW) January 8, 2010 16 / 31

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Minor operations Matroid deletion = graft contraction

M \ e ⇒ (G/e, T ′)

1 8 4 7 3 2 6 9 5

⇒

1 8 4 3 2 6 9 5

e = 7

Irene Pivotto (UW) January 8, 2010 17 / 31

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Minor operations

Representation (H, S) of N extends to M: M = ecut(G, T) ⇔ (G, T)

matroid major

⇑ ⇑

graft major

N = ecut(H, S) ⇔ (H, S)

Irene Pivotto (UW) January 8, 2010 18 / 31

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Finding excluded minors

N M

Irene Pivotto (UW) January 8, 2010 19 / 31

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Finding excluded minors

Idea: cover all the representations with equivalence classes.

Irene Pivotto (UW) January 8, 2010 20 / 31

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Finding excluded minors

Idea: cover all the representations with equivalence classes. Bundle: equivalence class of representations for some equivalence. N : even cut matroid Cover all the representations of N with bundles.

Irene Pivotto (UW) January 8, 2010 20 / 31

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Finding excluded minors

Idea: cover all the representations with equivalence classes. Bundle: equivalence class of representations for some equivalence. N : even cut matroid Cover all the representations of N with bundles. We want to study how the different bundles behave when taking majors. ⇒ stabilizer theorems (Whittle 1999).

Irene Pivotto (UW) January 8, 2010 20 / 31

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Stabilizer theorem

N, M even cut matroids, N ≤ M Bundle of

  • repr. of N

N M

Bundle of

  • repr. of M

Irene Pivotto (UW) January 8, 2010 21 / 31

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Stabilizer theorem

M N1 N

Irene Pivotto (UW) January 8, 2010 22 / 31

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Equivalence of grafts

Whitney flips: (G, T) ∼W (G ′, T ′) if

  • G ∼W G ′
  • every T-join of G is a T ′-join of G ′

G G'

Whitney flips preserve even cuts.

Irene Pivotto (UW) January 8, 2010 23 / 31

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Equivalence of grafts

Shuffle (Norine, Thomas) - preserves even cuts

a c d b a b c d a b c d a c d b

⇓

a' c' d' b' a' b' c' d' a' b' c' d' a c' d' b'

Irene Pivotto (UW) January 8, 2010 24 / 31

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Equivalence of grafts

Shuffle - example

1 8 10 7 9 6 14 13 12 11 15 5 4 3 2 9 2 7 10 1 6 14 13 15 11 12 5 4 3 8

Irene Pivotto (UW) January 8, 2010 25 / 31

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Bundles

Types of bundles: WS-bundles: related by ∼W , generated by a substantial graft WN-bundles: related by ∼W , generated by a non-substantial graft S-bundles: related by ∼S

Irene Pivotto (UW) January 8, 2010 26 / 31

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Bundles

(G, T) is non-substantial if, for some u, v ∈ V (G), (G, T △ {u, v}) ∼W (G ′, {u′, v′})

Irene Pivotto (UW) January 8, 2010 27 / 31

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Bundles

(G, T) is non-substantial if, for some u, v ∈ V (G), (G, T △ {u, v}) ∼W (G ′, {u′, v′}) Example

u v u' v'

Irene Pivotto (UW) January 8, 2010 27 / 31

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Bundles

Non-substantial graft do not extend uniquely.

u' v' u v

Irene Pivotto (UW) January 8, 2010 28 / 31

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Bundles

Non-substantial graft do not extend uniquely.

u' v' u v

But then they become substantial

Irene Pivotto (UW) January 8, 2010 28 / 31

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Bundles

Non-substantial graft do not extend uniquely.

u' v' u v

But then they become substantial ⇒ at most one branching

Irene Pivotto (UW) January 8, 2010 28 / 31

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Bundles

Non-substantial graft do not extend uniquely.

u' v' u v

But then they become substantial ⇒ at most one branching ⇒ at most twice the number of representations

Irene Pivotto (UW) January 8, 2010 28 / 31

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Bundles

... and then you prove a stabilizer theorem for each bundle...

Irene Pivotto (UW) January 8, 2010 29 / 31

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Bundles

... and then you prove a stabilizer theorem for each bundle... WS-bundles

√

WN-bundles

√

Irene Pivotto (UW) January 8, 2010 29 / 31

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Bundles

... and then you prove a stabilizer theorem for each bundle... WS-bundles

√

WN-bundles

√

S-bundles: in progress

Irene Pivotto (UW) January 8, 2010 29 / 31

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Future work

Escape theorems (how to kill the representations)

Irene Pivotto (UW) January 8, 2010 30 / 31

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Future work

Escape theorems (how to kill the representations) Excluded minors

Irene Pivotto (UW) January 8, 2010 30 / 31

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Future work

Escape theorems (how to kill the representations) Excluded minors Isomorphism theorem

Irene Pivotto (UW) January 8, 2010 30 / 31

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Future work

Escape theorems (how to kill the representations) Excluded minors Isomorphism theorem Parallel work in progress (joint work with B. Guenin and P. Wollan): Same approach for even cycle matroids

Irene Pivotto (UW) January 8, 2010 30 / 31

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Grazie Merci Thanks

Irene Pivotto (UW) January 8, 2010 31 / 31