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

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 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

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

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

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 4 5 1 The elements of cut( G ) are { 1 , 2 , . . . , 9 } 2 6 7 1 , 2 , 5 , 6 is a cycle of cut( G ) 3 8 9 Irene Pivotto (UW) January 8, 2010 4 / 31

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

Definitions Example 1 10 12 10 4 3 8 9 8 9 5 2 11 11 6 7 6 7 2 5 3 4 13 13 12 1 Irene Pivotto (UW) January 8, 2010 6 / 31

Definitions Whitney-flips preserve cuts: 1 10 12 10 4 3 9 9 8 8 6 2 7 11 11 5 6 2 7 5 3 4 13 13 12 1 Irene Pivotto (UW) January 8, 2010 7 / 31

Definitions Whitney-flips preserve cuts: 1 10 12 10 4 3 9 9 8 8 6 2 7 11 11 5 6 2 7 5 3 4 13 13 12 1 same cuts ⇔ related by ∼ W Irene Pivotto (UW) January 8, 2010 7 / 31

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

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

Definitions Example 4 5 1 2 6 7 3 8 9 Boxed vertices are in T Irene Pivotto (UW) January 8, 2010 9 / 31

Definitions Example 4 5 1 2 1 , 4 , 7 is an odd cut 6 7 3 8 9 Boxed vertices are in T Irene Pivotto (UW) January 8, 2010 10 / 31

Definitions Example 4 5 1 2 1 , 4 , 7 is an odd cut 6 7 1 , 2 , 5 , 6 is an even cut 3 8 9 Boxed vertices are in T Irene Pivotto (UW) January 8, 2010 11 / 31

Definitions Example 4 5 1 1 , 4 , 7 is an odd cut 2 6 7 1 , 2 , 5 , 6 is an even cut 3 1 , 4 , 7 , 6 , 3 , 8 is an even cut 8 9 Boxed vertices are in T Irene Pivotto (UW) January 8, 2010 12 / 31

Idea for finding excluded minors M : class of even cut matroids (a) Show: if M �∈ M , then M contains one of N 1 , . . . , N k (b) For all i , characterize matroids M minimally not in M with N i ≤ M . Irene Pivotto (UW) January 8, 2010 13 / 31

Idea for finding excluded minors M : class of even cut matroids (a) Show: if M �∈ M , then M contains one of N 1 , . . . , N k (b) For all i , characterize matroids M minimally not in M with N i ≤ M . √ (a) We can pick excluded minors for cographic matroids Irene Pivotto (UW) January 8, 2010 13 / 31

Idea for finding excluded minors M : class of even cut matroids (a) Show: if M �∈ M , then M contains one of N 1 , . . . , N k (b) For all i , characterize matroids M minimally not in M with N i ≤ 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

Many representations Example 1 11 2 9 11 8 6 6 12 15 12 15 8 9 2 1 10 7 7 10 5 5 3 3 14 13 14 13 4 4 Irene Pivotto (UW) January 8, 2010 14 / 31

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

Minor operations Matroid contraction = graft deletion M / e ⇒ ( G \ e , T ) 5 5 4 4 1 1 ⇒ 2 6 7 6 7 3 3 8 9 8 9 e = 2 Irene Pivotto (UW) January 8, 2010 16 / 31

Minor operations Matroid deletion = graft contraction M \ e ⇒ ( G / e , T ′ ) 5 5 4 4 1 1 ⇒ 2 2 6 7 6 3 3 8 9 8 9 e = 7 Irene Pivotto (UW) January 8, 2010 17 / 31

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

Finding excluded minors N M Irene Pivotto (UW) January 8, 2010 19 / 31

Finding excluded minors Idea: cover all the representations with equivalence classes. Irene Pivotto (UW) January 8, 2010 20 / 31

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

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

Stabilizer theorem N , M even cut matroids, N ≤ M N M Bundle of Bundle of repr. of N repr. of M Irene Pivotto (UW) January 8, 2010 21 / 31

Stabilizer theorem N M N 1 Irene Pivotto (UW) January 8, 2010 22 / 31

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

Equivalence of grafts Shuffle (Norine, Thomas) - preserves even cuts a b a b a b a b d c d c d c d c ⇓ a' b' a' b' a' b' a b' d' c' d' c' d' c' d' c' Irene Pivotto (UW) January 8, 2010 24 / 31

Equivalence of grafts Shuffle - example 1 11 2 9 11 8 6 6 12 15 12 15 8 9 2 1 10 7 7 10 5 5 3 3 14 13 14 13 4 4 Irene Pivotto (UW) January 8, 2010 25 / 31

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

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

Bundles ( G , T ) is non-substantial if, for some u , v ∈ V ( G ), ( G , T △ { u , v } ) ∼ W ( G ′ , { u ′ , v ′ } ) Example v u v' u' Irene Pivotto (UW) January 8, 2010 27 / 31

Bundles Non-substantial graft do not extend uniquely. v u' v' u Irene Pivotto (UW) January 8, 2010 28 / 31

Bundles Non-substantial graft do not extend uniquely. v u' v' u But then they become substantial Irene Pivotto (UW) January 8, 2010 28 / 31

Bundles Non-substantial graft do not extend uniquely. v u' v' u But then they become substantial ⇒ at most one branching Irene Pivotto (UW) January 8, 2010 28 / 31

Bundles Non-substantial graft do not extend uniquely. v u' v' u But then they become substantial ⇒ at most one branching ⇒ at most twice the number of representations Irene Pivotto (UW) January 8, 2010 28 / 31

Bundles ... and then you prove a stabilizer theorem for each bundle... Irene Pivotto (UW) January 8, 2010 29 / 31

Bundles ... and then you prove a stabilizer theorem for each bundle... √ WS-bundles √ WN-bundles Irene Pivotto (UW) January 8, 2010 29 / 31

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

Future work Escape theorems (how to kill the representations) Irene Pivotto (UW) January 8, 2010 30 / 31

Future work Escape theorems (how to kill the representations) Excluded minors Irene Pivotto (UW) January 8, 2010 30 / 31

Future work Escape theorems (how to kill the representations) Excluded minors Isomorphism theorem Irene Pivotto (UW) January 8, 2010 30 / 31

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