Oriented matroids and beyond Kolja Knauer Hans-J urgen Bandelt - - PowerPoint PPT Presentation

oriented matroids and beyond
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Oriented matroids and beyond Kolja Knauer Hans-J urgen Bandelt - - PowerPoint PPT Presentation

Oriented matroids and beyond Kolja Knauer Hans-J urgen Bandelt LIS, Aix-Marseille Universit e Universit at Hamburg Victor Chepoi LIS, Aix-Marseille Universit e Tilen Marc FMF, Univerza v Ljubljani S eminaire Francilien de G


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SLIDE 1

Oriented matroids and beyond

Kolja Knauer

LIS, Aix-Marseille Universit´ e

Hans-J¨ urgen Bandelt

Universit¨ at Hamburg

S´ eminaire Francilien de G´ eom´ etrie Algorithmique et Combinatoire F´ evrier 8, 2018 Victor Chepoi Tilen Marc

FMF, Univerza v Ljubljani LIS, Aix-Marseille Universit´ e

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SLIDE 2

Graphic oriented matroids

digraph D = (V, E)

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SLIDE 3

Graphic oriented matroids

digraph D = (V, E) minimal edge cut X

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SLIDE 4

Graphic oriented matroids

digraph D = (V, E) minimal edge cut X

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SLIDE 5

Graphic oriented matroids

digraph D = (V, E) minimal edge cut X + + − − −

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SLIDE 6

Graphic oriented matroids

digraph D = (V, E) minimal edge cut X + + − − − 0 0 0 00 0 0

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SLIDE 7

Graphic oriented matroids

digraph D = (V, E) minimal edge cut X + + − − − 0 0 0 00 0 0

               + . . . − − − +                ∈ {±, 0}E

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SLIDE 8

Graphic oriented matroids

digraph D = (V, E) minimal edge cut X + + − − − 0 0 0 00 0 0

               + . . . − − − +                ∈ {±, 0}E

cocircuit X ∈ C∗

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SLIDE 9

Graphic oriented matroids

digraph D = (V, E) minimal edge cut X 0 0 0 00 0 0

               + . . . − − − +                ∈ {±, 0}E

cocircuit X ∈ C∗ − − + + +

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SLIDE 10

Graphic oriented matroids

digraph D = (V, E) minimal edge cut X 0 0 0 00 0 0

               + . . . − − − +                ∈ {±, 0}E

cocircuit X ∈ C∗ − − + + +

               − . . . + + + −                ,

cocircuit −X ∈ C∗

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SLIDE 11

Graphic oriented matroids

digraph D = (V, E) minimal edge cut X 0 0 0 00 0 0

               + . . . − − − +                ∈ {±, 0}E

cocircuit X ∈ C∗ − − + + +

               − . . . + + + −                ,

cocircuit −X ∈ C∗ graphic oriented matroid of D: M = (E, C∗) ground set cocircuits

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SLIDE 12

From graphic to realizable

digraph D = (V, E) minimal edge cut X

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SLIDE 13

From graphic to realizable

digraph D = (V, E) minimal edge cut X incidence matrix I ∈ {±1, 0}V ×E a b c d e 1 2 3 4 5 6 7 8

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SLIDE 14

From graphic to realizable

digraph D = (V, E) minimal edge cut X incidence matrix I ∈ {±1, 0}V ×E                − . . . − + . . . + . . . + . . . + . . . + . . . − − . . . − . . . . . . . . . . . . . . . . . . ...                a b c d e 1 2 3 4 5 6 7 8 . . . . . . a b c d e 1 2 3 4 5 6 7 8

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SLIDE 15

From graphic to realizable

digraph D = (V, E) minimal edge cut X incidence matrix I ∈ {±1, 0}V ×E                − . . . − + . . . + . . . + . . . + . . . + . . . − − . . . − . . . . . . . . . . . . . . . . . . ...                a b c d e 1 2 3 4 5 6 7 8 . . . . . . a b c d e 1 2 3 4 5 6 7 8 +1· +1· +1· +1· −1· −1· −1· −1·

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SLIDE 16

From graphic to realizable

digraph D = (V, E) minimal edge cut X incidence matrix I ∈ {±1, 0}V ×E                − . . . − + . . . + . . . + . . . + . . . + . . . − − . . . − . . . . . . . . . . . . . . . . . . ...                a b c d e 1 2 3 4 5 6 7 8 . . . . . . a b c d e 1 2 3 4 5 6 7 8 +1· +1· +1· +1· −1· −1· −1· −1· Σ = ( −2 −2 +2 +2 +2 ? )

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SLIDE 17

From graphic to realizable

digraph D = (V, E) minimal edge cut X incidence matrix I ∈ {±1, 0}V ×E                − . . . − + . . . + . . . + . . . + . . . + . . . − − . . . − . . . . . . . . . . . . . . . . . . ...                a b c d e 1 2 3 4 5 6 7 8 . . . . . . a b c d e 1 2 3 4 5 6 7 8 +1· +1· +1· +1· −1· −1· −1· −1· Σ = ( −2 −2 +2 +2 +2 )

+1 −1

0 . . . 0

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SLIDE 18

From graphic to realizable

digraph D = (V, E) minimal edge cut X incidence matrix I ∈ {±1, 0}V ×E                − . . . − + . . . + . . . + . . . + . . . + . . . − − . . . − . . . . . . . . . . . . . . . . . . ...                a b c d e 1 2 3 4 5 6 7 8 . . . . . . a b c d e 1 2 3 4 5 6 7 8 +1· +1· +1· +1· −1· −1· −1· −1· Σ = ( −2 −2 +2 +2 +2 )

+1 −1

0 . . . 0 sgn(Σ) = − − + + + 0 . . . 0

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SLIDE 19

From graphic to realizable

digraph D = (V, E) minimal edge cut X incidence matrix I ∈ {±1, 0}V ×E                − . . . − + . . . + . . . + . . . + . . . + . . . − − . . . − . . . . . . . . . . . . . . . . . . ...                a b c d e 1 2 3 4 5 6 7 8 . . . . . . a b c d e 1 2 3 4 5 6 7 8 +1· +1· +1· +1· −1· −1· −1· −1· Σ = ( −2 −2 +2 +2 +2 )

+1 −1

0 . . . 0 sgn(Σ) = − − + + + 0 . . . 0

− − + + +

= X ∈ C∗

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SLIDE 20

From graphic to realizable

digraph D = (V, E) minimal edge cut X incidence matrix I ∈ {±1, 0}V ×E                − . . . − + . . . + . . . + . . . + . . . + . . . − − . . . − . . . . . . . . . . . . . . . . . . ...                a b c d e 1 2 3 4 5 6 7 8 . . . . . . a b c d e 1 2 3 4 5 6 7 8 +1· +1· +1· +1· −1· −1· −1· −1· Σ = ( −2 −2 +2 +2 +2 )

+1 −1

0 . . . 0 sgn(Σ) = − − + + + 0 . . . 0

− − + + +

= X ∈ C∗ C∗ := support-minimal sign vectors

  • f elements of row-space of I
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SLIDE 21

From graphic to realizable

digraph D = (V, E) minimal edge cut X incidence matrix I ∈ {±1, 0}V ×E                − . . . − + . . . + . . . + . . . + . . . + . . . − − . . . − . . . . . . . . . . . . . . . . . . ...                a b c d e 1 2 3 4 5 6 7 8 . . . . . . a b c d e 1 2 3 4 5 6 7 8 +1· +1· +1· +1· −1· −1· −1· −1· Σ = ( −2 −2 +2 +2 +2 )

+1 −1

0 . . . 0 sgn(Σ) = − − + + + 0 . . . 0

− − + + +

= X ∈ C∗ C∗ := support-minimal sign vectors

  • f elements of row-space of I
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SLIDE 22

From graphic to realizable

digraph D = (V, E) minimal edge cut X incidence matrix I ∈ {±1, 0}V ×E                − . . . − + . . . + . . . + . . . + . . . + . . . − − . . . − . . . . . . . . . . . . . . . . . . ...                a b c d e 1 2 3 4 5 6 7 8 . . . . . . a b c d e 1 2 3 4 5 6 7 8 +1· +1· +1· +1· −1· −1· −1· −1· Σ = ( −2 −2 +2 +2 +2 )

+1 −1

0 . . . 0 sgn(Σ) = − − + + + 0 . . . 0

− − + + +

= X ∈ C∗ C∗ := support-minimal sign vectors

  • f elements of row-space of I

a b c d e

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SLIDE 23

From graphic to realizable

digraph D = (V, E) minimal edge cut X incidence matrix I ∈ {±1, 0}V ×E                − . . . − + . . . + . . . + . . . + . . . + . . . − − . . . − . . . . . . . . . . . . . . . . . . ...                a b c d e 1 2 3 4 5 6 7 8 . . . . . . a b c d e 1 2 3 4 5 6 7 8 +1· +1· +1· +1· −1· −1· −1· −1· Σ = ( −2 −2 +2 +2 +2 )

+1 −1

0 . . . 0 sgn(Σ) = − − + + + 0 . . . 0

− − + + +

= X ∈ C∗ C∗ := support-minimal sign vectors

  • f elements of row-space of I

a b c d e X

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SLIDE 24

From graphic to realizable

digraph D = (V, E) minimal edge cut X incidence matrix I ∈ {±1, 0}V ×E                − . . . − + . . . + . . . + . . . + . . . + . . . − − . . . − . . . . . . . . . . . . . . . . . . ...                a b c d e 1 2 3 4 5 6 7 8 . . . . . . a b c d e 1 2 3 4 5 6 7 8 +1· +1· +1· +1· −1· −1· −1· −1· Σ = ( −2 −2 +2 +2 +2 )

+1 −1

0 . . . 0 sgn(Σ) = − − + + + 0 . . . 0

− − + + +

= X ∈ C∗ C∗ := support-minimal sign vectors

  • f elements of row-space of I

a b c d e X sign vectors of min-dimensional cells of hyperplane arrangement

=

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SLIDE 25

Acyclic orientations

digraph D = (V, E) minimal edge cut X incidence matrix I ∈ {±1, 0}V ×E                − . . . − + . . . + . . . + . . . + . . . + . . . − − . . . − . . . . . . . . . . . . . . . . . . ...                a b c d e 1 2 3 4 5 6 7 8 . . . . . . a b c d e 1 2 3 4 5 6 7 8 +1· +1· +1· +1· −1· −1· −1· −1·

+1 −1

sgn(Σ) = − − + + + 0 . . . 0 = X

− − + + +

a b c d e X

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SLIDE 26

Acyclic orientations

digraph D = (V, E) minimal edge cut X incidence matrix I ∈ {±1, 0}V ×E                − . . . − + . . . + . . . + . . . + . . . + . . . − − . . . − . . . . . . . . . . . . . . . . . . ...                a b c d e 1 2 3 4 5 6 7 8 . . . . . . a b c d e 1 2 3 4 5 6 7 8 +1· +1· +1· +1· −1· −1· −1· −1·

+1 −1

sgn(Σ) = − − + + + 0 . . . 0 = X

− − + + +

a b c d e X

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SLIDE 27

Acyclic orientations

digraph D = (V, E) minimal edge cut X incidence matrix I ∈ {±1, 0}V ×E                − . . . − + . . . + . . . + . . . + . . . + . . . − − . . . − . . . . . . . . . . . . . . . . . . ...                a b c d e 1 2 3 4 5 6 7 8 . . . . . . a b c d e 1 2 3 4 5 6 7 8 +1· +1· +1· +1· −1· −1· −1· −1·

+1 −1

sgn(Σ) = − − + + + 0 . . . 0 = X

− − + + +

a b c d e X + −

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SLIDE 28

Acyclic orientations

digraph D = (V, E) minimal edge cut X incidence matrix I ∈ {±1, 0}V ×E                − . . . − + . . . + . . . + . . . + . . . + . . . − − . . . − . . . . . . . . . . . . . . . . . . ...                a b c d e 1 2 3 4 5 6 7 8 . . . . . . a b c d e 1 2 3 4 5 6 7 8 +1· +1· +1· +1· −1· −1· −1· −1·

+1 −1

sgn(Σ) = − − + + + 0 . . . 0 = X

− − + + +

a b c d e X + − +

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SLIDE 29

Acyclic orientations

digraph D = (V, E) minimal edge cut X incidence matrix I ∈ {±1, 0}V ×E                − . . . − + . . . + . . . + . . . + . . . + . . . − − . . . − . . . . . . . . . . . . . . . . . . ...                a b c d e 1 2 3 4 5 6 7 8 . . . . . . a b c d e 1 2 3 4 5 6 7 8 +1· +1· +1· +1· −1· −1· −1· −1·

+1 −1

sgn(Σ) = − − + + + 0 . . . 0 = X

− − + + +

a b c d e X + − +

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SLIDE 30

Acyclic orientations

digraph D = (V, E) minimal edge cut X incidence matrix I ∈ {±1, 0}V ×E                − . . . − + . . . + . . . + . . . + . . . + . . . − − . . . − . . . . . . . . . . . . . . . . . . ...                a b c d e 1 2 3 4 5 6 7 8 . . . . . . a b c d e 1 2 3 4 5 6 7 8 +1· +1· +1· +1· −1· −1· −1· −1·

+1 −1

sgn(Σ) = − − + + + 0 . . . 0 = X

− − + + +

a b c d e X + − + reorientation

slide-31
SLIDE 31

Acyclic orientations

digraph D = (V, E) minimal edge cut X incidence matrix I ∈ {±1, 0}V ×E                − . . . − + . . . + . . . + . . . + . . . + . . . − − . . . − . . . . . . . . . . . . . . . . . . ...                a b c d e 1 2 3 4 5 6 7 8 . . . . . . a b c d e 1 2 3 4 5 6 7 8 +1· +1· +1· +1· −1· −1· −1· −1·

+1 −1

sgn(Σ) = − − + + + 0 . . . 0 = X

− − + + +

a b c d e X + − + reorientation

slide-32
SLIDE 32

Acyclic orientations

digraph D = (V, E) minimal edge cut X incidence matrix I ∈ {±1, 0}V ×E                − . . . − + . . . + . . . + . . . + . . . + . . . − − . . . − . . . . . . . . . . . . . . . . . . ...                a b c d e 1 2 3 4 5 6 7 8 . . . . . . a b c d e 1 2 3 4 5 6 7 8 +1· +1· +1· +1· −1· −1· −1· −1·

+1 −1

sgn(Σ) = − − + + + 0 . . . 0 = X

− − + + +

a b c d e X + − + reorientation + − +

slide-33
SLIDE 33

Acyclic orientations

digraph D = (V, E) minimal edge cut X incidence matrix I ∈ {±1, 0}V ×E                − . . . − + . . . + . . . + . . . + . . . + . . . − − . . . − . . . . . . . . . . . . . . . . . . ...                a b c d e 1 2 3 4 5 6 7 8 . . . . . . a b c d e 1 2 3 4 5 6 7 8 +1· +1· +1· +1· −1· −1· −1· −1·

+1 −1

sgn(Σ) = − − + + + 0 . . . 0 = X

− − + + +

a b c d e X + − + reorientation + − + directed

slide-34
SLIDE 34

Acyclic orientations

digraph D = (V, E) minimal edge cut X incidence matrix I ∈ {±1, 0}V ×E                − . . . − + . . . + . . . + . . . + . . . + . . . − − . . . − . . . . . . . . . . . . . . . . . . ...                a b c d e 1 2 3 4 5 6 7 8 . . . . . . a b c d e 1 2 3 4 5 6 7 8 +1· +1· +1· +1· −1· −1· −1· −1·

+1 −1

sgn(Σ) = − − + + + 0 . . . 0 = X

− − + + +

a b c d e X + − + reorientation + − + directed every edge in a directed cut

slide-35
SLIDE 35

Acyclic orientations

digraph D = (V, E) minimal edge cut X incidence matrix I ∈ {±1, 0}V ×E                − . . . − + . . . + . . . + . . . + . . . + . . . − − . . . − . . . . . . . . . . . . . . . . . . ...                a b c d e 1 2 3 4 5 6 7 8 . . . . . . a b c d e 1 2 3 4 5 6 7 8 +1· +1· +1· +1· −1· −1· −1· −1·

+1 −1

sgn(Σ) = − − + + + 0 . . . 0 = X

− − + + +

a b c d e X + − + reorientation + − + directed every edge in a directed cut

slide-36
SLIDE 36

Acyclic orientations

digraph D = (V, E) minimal edge cut X incidence matrix I ∈ {±1, 0}V ×E                − . . . − + . . . + . . . + . . . + . . . + . . . − − . . . − . . . . . . . . . . . . . . . . . . ...                a b c d e 1 2 3 4 5 6 7 8 . . . . . . a b c d e 1 2 3 4 5 6 7 8 +1· +1· +1· +1· −1· −1· −1· −1·

+1 −1

sgn(Σ) = − − + + + 0 . . . 0 = X

− − + + +

a b c d e X + − + reorientation + − + directed every edge in a directed cut

slide-37
SLIDE 37

Acyclic orientations

digraph D = (V, E) minimal edge cut X incidence matrix I ∈ {±1, 0}V ×E                − . . . − + . . . + . . . + . . . + . . . + . . . − − . . . − . . . . . . . . . . . . . . . . . . ...                a b c d e 1 2 3 4 5 6 7 8 . . . . . . a b c d e 1 2 3 4 5 6 7 8 +1· +1· +1· +1· −1· −1· −1· −1·

+1 −1

sgn(Σ) = − − + + + 0 . . . 0 = X

− − + + +

a b c d e X + − + reorientation + − + directed every edge in a directed cut ⇐ ⇒ acyclic orientation

slide-38
SLIDE 38

Acyclic orientations

digraph D = (V, E) minimal edge cut X incidence matrix I ∈ {±1, 0}V ×E                − . . . − + . . . + . . . + . . . + . . . + . . . − − . . . − . . . . . . . . . . . . . . . . . . ...                a b c d e 1 2 3 4 5 6 7 8 . . . . . . a b c d e 1 2 3 4 5 6 7 8 +1· +1· +1· +1· −1· −1· −1· −1·

+1 −1

sgn(Σ) = − − + + + 0 . . . 0 = X

− − + + +

a b c d e X + − + reorientation + − + directed every edge in a directed cut ⇐ ⇒ acyclic orientation max-dimensional cells ∼ = acyclic orientations

slide-39
SLIDE 39

Acyclic orientations

digraph D = (V, E) minimal edge cut X incidence matrix I ∈ {±1, 0}V ×E                − . . . − + . . . + . . . + . . . + . . . + . . . − − . . . − . . . . . . . . . . . . . . . . . . ...                a b c d e 1 2 3 4 5 6 7 8 . . . . . . a b c d e 1 2 3 4 5 6 7 8 +1· +1· +1· +1· −1· −1· −1· −1·

+1 −1

sgn(Σ) = − − + + + 0 . . . 0 = X

− − + + +

a b c d e X + − + reorientation + − + directed every edge in a directed cut ⇐ ⇒ acyclic orientation max-dimensional cells ∼ = acyclic orientations

slide-40
SLIDE 40

Acyclic orientations

digraph D = (V, E) minimal edge cut X incidence matrix I ∈ {±1, 0}V ×E                − . . . − + . . . + . . . + . . . + . . . + . . . − − . . . − . . . . . . . . . . . . . . . . . . ...                a b c d e 1 2 3 4 5 6 7 8 . . . . . . a b c d e 1 2 3 4 5 6 7 8 +1· +1· +1· +1· −1· −1· −1· −1·

+1 −1

sgn(Σ) = − − + + + 0 . . . 0 = X

− − + + +

a b c d e X + − + reorientation + − + directed every edge in a directed cut ⇐ ⇒ acyclic orientation max-dimensional cells ∼ = acyclic orientations

slide-41
SLIDE 41

Acyclic orientations

digraph D = (V, E) minimal edge cut X incidence matrix I ∈ {±1, 0}V ×E                − . . . − + . . . + . . . + . . . + . . . + . . . − − . . . − . . . . . . . . . . . . . . . . . . ...                a b c d e 1 2 3 4 5 6 7 8 . . . . . . a b c d e 1 2 3 4 5 6 7 8 +1· +1· +1· +1· −1· −1· −1· −1·

+1 −1

sgn(Σ) = − − + + + 0 . . . 0 = X

− − + + +

a b c d e X + − + reorientation + − + directed every edge in a directed cut ⇐ ⇒ acyclic orientation max-dimensional cells ∼ = acyclic orientations flip-graph on acyclic

  • rientations ∼

= region graph of arrangement

slide-42
SLIDE 42

Realizable oriented matroids

C∗ := support-minimal sign vectors

  • f elements of R-vector space = sign vectors of min-dimensional cells
  • f (central) hyperplane arrangement
slide-43
SLIDE 43

Realizable oriented matroids

C∗ := support-minimal sign vectors

  • f elements of R-vector space = sign vectors of min-dimensional cells
  • f (central) hyperplane arrangement
slide-44
SLIDE 44

Realizable oriented matroids

C∗ := support-minimal sign vectors

  • f elements of R-vector space = sign vectors of min-dimensional cells
  • f (central) hyperplane arrangement
slide-45
SLIDE 45

Realizable oriented matroids

C∗ := support-minimal sign vectors

  • f elements of R-vector space =

sign vectors of 0-dimensional cells of arrangement of great cycles on sphere

=

sign vectors of min-dimensional cells

  • f (central) hyperplane arrangement
slide-46
SLIDE 46

Thm[Folkman, Lawrence ’78] correspondence of pseudo-sphere arrangements and oriented matroids.

Oriented matroids

slide-47
SLIDE 47

Thm[Folkman, Lawrence ’78] correspondence of pseudo-sphere arrangements and oriented matroids.

Oriented matroids

slide-48
SLIDE 48

0 + −0

Thm[Folkman, Lawrence ’78] correspondence of pseudo-sphere arrangements and oriented matroids.

Oriented matroids

slide-49
SLIDE 49

+ − −+ 0 + −0

Thm[Folkman, Lawrence ’78] correspondence of pseudo-sphere arrangements and oriented matroids.

Oriented matroids

slide-50
SLIDE 50

+ − −+ 0 + −0

Thm[Folkman, Lawrence ’78] correspondence of pseudo-sphere arrangements and oriented matroids.

Oriented matroids

+0 − +

slide-51
SLIDE 51

+ − −+ 0 + −0

Thm[Folkman, Lawrence ’78] correspondence of pseudo-sphere arrangements and oriented matroids.

  • Covector axioms: (E, L) OM iff

(Z) 0 ∈ L (FS) L ◦ −L ⊆ L (SE) ∀X, Y ∈ L and e ∈ S(X, Y )∃Z ∈ L : Ze = 0 and Zf = Xf ◦ Yf for f / ∈ S(X, Y ).

Oriented matroids

+0 − +

slide-52
SLIDE 52

+ − −+

topes T of M= maximal cells=

  • max. covectors

Thm[Folkman, Lawrence ’78] correspondence of pseudo-sphere arrangements and oriented matroids.

  • Covector axioms: (E, L) OM iff

(Z) 0 ∈ L (FS) L ◦ −L ⊆ L (SE) ∀X, Y ∈ L and e ∈ S(X, Y )∃Z ∈ L : Ze = 0 and Zf = Xf ◦ Yf for f / ∈ S(X, Y ).

Oriented matroids

slide-53
SLIDE 53

+ − −+

topes T of M= maximal cells=

  • max. covectors

Thm[Folkman, Lawrence ’78] correspondence of pseudo-sphere arrangements and oriented matroids.

  • Covector axioms: (E, L) OM iff

(Z) 0 ∈ L (FS) L ◦ −L ⊆ L (SE) ∀X, Y ∈ L and e ∈ S(X, Y )∃Z ∈ L : Ze = 0 and Zf = Xf ◦ Yf for f / ∈ S(X, Y ).

Oriented matroids

slide-54
SLIDE 54

+ − −+

tope graph GL =incidence graph= induced graph in QE topes T of M= maximal cells=

  • max. covectors

Thm[Folkman, Lawrence ’78] correspondence of pseudo-sphere arrangements and oriented matroids.

  • Covector axioms: (E, L) OM iff

(Z) 0 ∈ L (FS) L ◦ −L ⊆ L (SE) ∀X, Y ∈ L and e ∈ S(X, Y )∃Z ∈ L : Ze = 0 and Zf = Xf ◦ Yf for f / ∈ S(X, Y ).

Oriented matroids

slide-55
SLIDE 55

Thm[Karlander ’92] correspondence between affine arrangements of pseudospheres and affine oriented matroids.

+ − −+ 0 + −0

Affine oriented matroids

slide-56
SLIDE 56

Thm[Karlander ’92] correspondence between affine arrangements of pseudospheres and affine oriented matroids.

+ − −+ 0 + −0

Affine oriented matroids

slide-57
SLIDE 57

Thm[Karlander ’92] correspondence between affine arrangements of pseudospheres and affine oriented matroids.

+ − −+ 0 + −0

  • Covector axioms: (E, L) affine oriented matroid:

(A) something lengthy (FS) L ◦ −L ⊆ L (SE) ∀X, Y ∈ L and e ∈ S(X, Y )∃Z ∈ L : Ze = 0 and Zf = Xf ◦ Yf for f / ∈ S(X, Y ).

Affine oriented matroids

slide-58
SLIDE 58

Thm[Karlander ’92] correspondence between affine arrangements of pseudospheres and affine oriented matroids.

+ − −+ 0 + −0

  • Covector axioms: (E, L) affine oriented matroid:

(A) something lengthy (FS) L ◦ −L ⊆ L (SE) ∀X, Y ∈ L and e ∈ S(X, Y )∃Z ∈ L : Ze = 0 and Zf = Xf ◦ Yf for f / ∈ S(X, Y ). tope graph GL =incidence graph= induced graph in QE

Affine oriented matroids

topes T of M= maximal cells=

  • max. covectors
slide-59
SLIDE 59

Thm[Karlander ’92] correspondence between affine arrangements of pseudospheres and affine oriented matroids.

+ − −+ 0 + −0

  • Covector axioms: (E, L) affine oriented matroid:

(A) something lengthy (FS) L ◦ −L ⊆ L (SE) ∀X, Y ∈ L and e ∈ S(X, Y )∃Z ∈ L : Ze = 0 and Zf = Xf ◦ Yf for f / ∈ S(X, Y ). tope graph GL =incidence graph= induced graph in QE

Affine oriented matroids

topes T of M= maximal cells=

  • max. covectors

digraph example: topes T ∼ = acyclic orientations with edge e’s orientation fixed

e

slide-60
SLIDE 60

Thm[Karlander ’92] correspondence between affine arrangements of pseudospheres and affine oriented matroids.

+ − −+ 0 + −0

  • Covector axioms: (E, L) affine oriented matroid:

(A) something lengthy (FS) L ◦ −L ⊆ L (SE) ∀X, Y ∈ L and e ∈ S(X, Y )∃Z ∈ L : Ze = 0 and Zf = Xf ◦ Yf for f / ∈ S(X, Y ). tope graph GL =incidence graph= induced graph in QE

Affine oriented matroids

topes T of M= maximal cells=

  • max. covectors

digraph example: topes T ∼ = acyclic orientations with edge e’s orientation fixed

e why not fix more?

Bandelt, Chepoi, K ’15:

slide-61
SLIDE 61

Complexes of oriented matroids

tope graph GL =incidence graph= induced graph in QE topes T of M= maximal cells=

  • max. covectors
  • Covector axioms: (E, L) COM iff

(FS) L ◦ −L ⊆ L (SE) ∀X, Y ∈ L and e ∈ S(X, Y )∃Z ∈ L : Ze = 0 and Zf = Xf ◦ Yf for f / ∈ S(X, Y ).

0 + −0 + − −+

why not fix more?

Bandelt, Chepoi, K ’15:

slide-62
SLIDE 62

Def[Bandelt, Chepoi, K ’15] realizable COM = sign systems from arrangement of open halfspaces and hyperplanes.

  • Covector axioms: (E, L) COM iff

(FS) L ◦ −L ⊆ L (SE) ∀X, Y ∈ L and e ∈ S(X, Y )∃Z ∈ L : Ze = 0 and Zf = Xf ◦ Yf for f / ∈ S(X, Y ). tope graph GL =incidence graph= induced graph in QE topes T of M= maximal cells=

  • max. covectors

Complexes of oriented matroids

slide-63
SLIDE 63

Def[Bandelt, Chepoi, K ’15] realizable COM = sign systems from arrangement of open halfspaces and hyperplanes.

  • Covector axioms: (E, L) COM iff

(FS) L ◦ −L ⊆ L (SE) ∀X, Y ∈ L and e ∈ S(X, Y )∃Z ∈ L : Ze = 0 and Zf = Xf ◦ Yf for f / ∈ S(X, Y ). tope graph GL =incidence graph= induced graph in QE topes T of M= maximal cells=

  • max. covectors

    + − +     ◦ (−     + + + +    ) =     − + − +    

(FS)

Complexes of oriented matroids

slide-64
SLIDE 64

Def[Bandelt, Chepoi, K ’15] realizable COM = sign systems from arrangement of open halfspaces and hyperplanes.

  • Covector axioms: (E, L) COM iff

(FS) L ◦ −L ⊆ L (SE) ∀X, Y ∈ L and e ∈ S(X, Y )∃Z ∈ L : Ze = 0 and Zf = Xf ◦ Yf for f / ∈ S(X, Y ). tope graph GL =incidence graph= induced graph in QE topes T of M= maximal cells=

  • max. covectors

(FS)

Complexes of oriented matroids

    + − +     ◦     − − − −     =     − + − +    

slide-65
SLIDE 65

Def[Bandelt, Chepoi, K ’15] realizable COM = sign systems from arrangement of open halfspaces and hyperplanes.

  • Covector axioms: (E, L) COM iff

(FS) L ◦ −L ⊆ L (SE) ∀X, Y ∈ L and e ∈ S(X, Y )∃Z ∈ L : Ze = 0 and Zf = Xf ◦ Yf for f / ∈ S(X, Y ). X Y tope graph GL =incidence graph= induced graph in QE topes T of M= maximal cells=

  • max. covectors

(FS)

Complexes of oriented matroids

    + − +     ◦     − − − −     =     − + − +    

slide-66
SLIDE 66

Def[Bandelt, Chepoi, K ’15] realizable COM = sign systems from arrangement of open halfspaces and hyperplanes.

  • Covector axioms: (E, L) COM iff

(FS) L ◦ −L ⊆ L (SE) ∀X, Y ∈ L and e ∈ S(X, Y )∃Z ∈ L : Ze = 0 and Zf = Xf ◦ Yf for f / ∈ S(X, Y ). X Y tope graph GL =incidence graph= induced graph in QE topes T of M= maximal cells=

  • max. covectors

(FS)

Complexes of oriented matroids

    + − +     ◦     − − − −     =     − + − +    

slide-67
SLIDE 67

Def[Bandelt, Chepoi, K ’15] realizable COM = sign systems from arrangement of open halfspaces and hyperplanes.

  • Covector axioms: (E, L) COM iff

(FS) L ◦ −L ⊆ L (SE) ∀X, Y ∈ L and e ∈ S(X, Y )∃Z ∈ L : Ze = 0 and Zf = Xf ◦ Yf for f / ∈ S(X, Y ). X Y X ◦ −Y tope graph GL =incidence graph= induced graph in QE topes T of M= maximal cells=

  • max. covectors

(FS)

Complexes of oriented matroids

    + − +     ◦     − − − −     =     − + − +    

slide-68
SLIDE 68

Def[Bandelt, Chepoi, K ’15] realizable COM = sign systems from arrangement of open halfspaces and hyperplanes.

  • Covector axioms: (E, L) COM iff

(FS) L ◦ −L ⊆ L (SE) ∀X, Y ∈ L and e ∈ S(X, Y )∃Z ∈ L : Ze = 0 and Zf = Xf ◦ Yf for f / ∈ S(X, Y ). tope graph GL =incidence graph= induced graph in QE topes T of M= maximal cells=

  • max. covectors

    + − +     ,     − − − −    

Complexes of oriented matroids

(SE)

slide-69
SLIDE 69

Def[Bandelt, Chepoi, K ’15] realizable COM = sign systems from arrangement of open halfspaces and hyperplanes.

  • Covector axioms: (E, L) COM iff

(FS) L ◦ −L ⊆ L (SE) ∀X, Y ∈ L and e ∈ S(X, Y )∃Z ∈ L : Ze = 0 and Zf = Xf ◦ Yf for f / ∈ S(X, Y ). tope graph GL =incidence graph= induced graph in QE topes T of M= maximal cells=

  • max. covectors

    + − +     ,     − − − −    

e

Complexes of oriented matroids

(SE)

slide-70
SLIDE 70

Def[Bandelt, Chepoi, K ’15] realizable COM = sign systems from arrangement of open halfspaces and hyperplanes.

  • Covector axioms: (E, L) COM iff

(FS) L ◦ −L ⊆ L (SE) ∀X, Y ∈ L and e ∈ S(X, Y )∃Z ∈ L : Ze = 0 and Zf = Xf ◦ Yf for f / ∈ S(X, Y ). tope graph GL =incidence graph= induced graph in QE topes T of M= maximal cells=

  • max. covectors

    + − +     ,     − − − −    

   − − ?    

e

Complexes of oriented matroids

(SE)

slide-71
SLIDE 71

Def[Bandelt, Chepoi, K ’15] realizable COM = sign systems from arrangement of open halfspaces and hyperplanes.

  • Covector axioms: (E, L) COM iff

(FS) L ◦ −L ⊆ L (SE) ∀X, Y ∈ L and e ∈ S(X, Y )∃Z ∈ L : Ze = 0 and Zf = Xf ◦ Yf for f / ∈ S(X, Y ). X Y e tope graph GL =incidence graph= induced graph in QE topes T of M= maximal cells=

  • max. covectors

    + − +     ,     − − − −    

   − − ?    

e

Complexes of oriented matroids

(SE)

slide-72
SLIDE 72

Def[Bandelt, Chepoi, K ’15] realizable COM = sign systems from arrangement of open halfspaces and hyperplanes.

  • Covector axioms: (E, L) COM iff

(FS) L ◦ −L ⊆ L (SE) ∀X, Y ∈ L and e ∈ S(X, Y )∃Z ∈ L : Ze = 0 and Zf = Xf ◦ Yf for f / ∈ S(X, Y ). X Y e Z tope graph GL =incidence graph= induced graph in QE topes T of M= maximal cells=

  • max. covectors

    + − +     ,     − − − −    

   − − ?    

e

Complexes of oriented matroids

(SE)

slide-73
SLIDE 73

Def[Bandelt, Chepoi, K ’15] realizable COM = sign systems from arrangement of open halfspaces and hyperplanes.

  • Covector axioms: (E, L) COM iff

(FS) L ◦ −L ⊆ L (SE) ∀X, Y ∈ L and e ∈ S(X, Y )∃Z ∈ L : Ze = 0 and Zf = Xf ◦ Yf for f / ∈ S(X, Y ). tope graph GL =incidence graph= induced graph in QE topes T of M= maximal cells=

  • max. covectors

Complexes of oriented matroids

digraph example: topes T ∼ = acyclic orientations with edges E’s orientation fixed e1 e2 e3 e4 e5 e6 e7 e8 e9

slide-74
SLIDE 74

Def[Bandelt, Chepoi, K ’15] realizable COM = sign systems from arrangement of open halfspaces and hyperplanes.

  • Covector axioms: (E, L) COM iff

(FS) L ◦ −L ⊆ L (SE) ∀X, Y ∈ L and e ∈ S(X, Y )∃Z ∈ L : Ze = 0 and Zf = Xf ◦ Yf for f / ∈ S(X, Y ). tope graph GL =incidence graph= induced graph in QE topes T of M= maximal cells=

  • max. covectors

Complexes of oriented matroids

topes T ∼ = acyclic orientations

  • f a mixed graph
slide-75
SLIDE 75
  • Covector axioms: (E, L) oriented matroid:

(Z) ∅ ∈ L (FS) L ◦ −L ⊆ L (SE) ∀X, Y ∈ L and e ∈ S(X, Y )∃Z ∈ L : Ze = 0 and Zf = Xf ◦ Yf for f / ∈ S(X, Y ).

  • Covector axioms: (E, L) affine oriented matroid:

(A) something lengthy (FS) L ◦ −L ⊆ L (SE) ∀X, Y ∈ L and e ∈ S(X, Y )∃Z ∈ L : Ze = 0 and Zf = Xf ◦ Yf for f / ∈ S(X, Y ).

  • Covector axioms: (E, L) COM iff

(FS) L ◦ −L ⊆ L (SE) ∀X, Y ∈ L and e ∈ S(X, Y )∃Z ∈ L : Ze = 0 and Zf = Xf ◦ Yf for f / ∈ S(X, Y ).

A common generalization

slide-76
SLIDE 76
  • Covector axioms: (E, L) oriented matroid:

(Z) ∅ ∈ L (FS) L ◦ −L ⊆ L (SE) ∀X, Y ∈ L and e ∈ S(X, Y )∃Z ∈ L : Ze = 0 and Zf = Xf ◦ Yf for f / ∈ S(X, Y ).

  • Covector axioms: (E, L) affine oriented matroid:

(A) something lengthy (FS) L ◦ −L ⊆ L (SE) ∀X, Y ∈ L and e ∈ S(X, Y )∃Z ∈ L : Ze = 0 and Zf = Xf ◦ Yf for f / ∈ S(X, Y ).

  • Covector axioms: (E, L) COM iff

(FS) L ◦ −L ⊆ L (SE) ∀X, Y ∈ L and e ∈ S(X, Y )∃Z ∈ L : Ze = 0 and Zf = Xf ◦ Yf for f / ∈ S(X, Y ).

t

  • p

e g r a p h s p a r t i a l c u b e s a n d d e t e r m i n e L

A common generalization

slide-77
SLIDE 77

G partial cube :⇔ G isometric subgraph of hypercube G ⊆ Qn such that dG(v, w) = dQn(v, w)∀v, w ∈ G

Partial cubes and partial cube minors

slide-78
SLIDE 78

G partial cube :⇔ G isometric subgraph of hypercube tope graph of realizable COM (arrangement of half and hyperplanes) G ⊆ Qn such that dG(v, w) = dQn(v, w)∀v, w ∈ G

Partial cubes and partial cube minors

slide-79
SLIDE 79

G partial cube :⇔ G isometric subgraph of hypercube tope graph of realizable COM (arrangement of half and hyperplanes) G ⊆ Qn such that dG(v, w) = dQn(v, w)∀v, w ∈ G

Partial cubes and partial cube minors

slide-80
SLIDE 80

G partial cube :⇔ G isometric subgraph of hypercube tope graph of realizable COM (arrangement of half and hyperplanes) G ⊆ Qn such that dG(v, w) = dQn(v, w)∀v, w ∈ G

Partial cubes and partial cube minors

slide-81
SLIDE 81

G partial cube :⇔ G isometric subgraph of hypercube edges of partial cube naturally partitioned into minimal cuts C G ⊆ Qn such that dG(v, w) = dQn(v, w)∀v, w ∈ G

Partial cubes and partial cube minors

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SLIDE 82

G partial cube :⇔ G isometric subgraph of hypercube edges of partial cube naturally partitioned into minimal cuts C minor-relation G ⊆ Qn such that dG(v, w) = dQn(v, w)∀v, w ∈ G

Partial cubes and partial cube minors

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SLIDE 83

G partial cube :⇔ G isometric subgraph of hypercube edges of partial cube naturally partitioned into minimal cuts C restriction to a side of a cut minor-relation G ⊆ Qn such that dG(v, w) = dQn(v, w)∀v, w ∈ G

Partial cubes and partial cube minors

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SLIDE 84

G partial cube :⇔ G isometric subgraph of hypercube edges of partial cube naturally partitioned into minimal cuts C restriction to a side of a cut contraction of a cut minor-relation G ⊆ Qn such that dG(v, w) = dQn(v, w)∀v, w ∈ G

Partial cubes and partial cube minors

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SLIDE 85

G partial cube :⇔ G isometric subgraph of hypercube edges of partial cube naturally partitioned into minimal cuts C restriction to a side of a cut contraction of a cut minor-relation yields new partial cube G ⊆ Qn such that dG(v, w) = dQn(v, w)∀v, w ∈ G

Partial cubes and partial cube minors

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SLIDE 86

G partial cube :⇔ G isometric subgraph of hypercube edges of partial cube naturally partitioned into minimal cuts C minor-relation tope graph of realizable COM (arrangement of half and hyperplanes) yields new tope graph G ⊆ Qn such that dG(v, w) = dQn(v, w)∀v, w ∈ G

Partial cubes and partial cube minors

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SLIDE 87

some minor-closed classes planar partial cubes GCOM graphs of arrangements

  • f half- and hyperplanes

Hypercellular graphs partial cubes

Partial cube minors

median graphs graphs of acyclic

  • rs of mixed graphs

distributive lattices bipartite cellular graphs

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SLIDE 88

some minor-closed classes planar partial cubes GCOM graphs of arrangements

  • f half- and hyperplanes

Hypercellular graphs partial cubes each has a family of excluded minors

Partial cube minors

median graphs graphs of acyclic

  • rs of mixed graphs

distributive lattices bipartite cellular graphs

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SLIDE 89

some minor-closed classes planar partial cubes GCOM graphs of arrangements

  • f half- and hyperplanes

Hypercellular graphs partial cubes each has a family of excluded minors F(Q−) =

Partial cube minors

median graphs graphs of acyclic

  • rs of mixed graphs

distributive lattices Thm[K, Marc] bipartite cellular graphs

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SLIDE 90

some minor-closed classes planar partial cubes GCOM graphs of arrangements

  • f half- and hyperplanes

Hypercellular graphs partial cubes each has a family of excluded minors F(Q−) = F( )

Partial cube minors

median graphs

=

graphs of acyclic

  • rs of mixed graphs

F( ,

=

) F( ,

=

) F( ,

=

) distributive lattices , Thm[K, Marc] Thm[Chepoi, K, Marc] bipartite cellular graphs

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SLIDE 91

some minor-closed classes planar partial cubes GCOM graphs of arrangements

  • f half- and hyperplanes

Hypercellular graphs partial cubes each has a family of excluded minors F(Q−) = F( ) F(?)

=

F(?)

=

Partial cube minors

median graphs

=

graphs of acyclic

  • rs of mixed graphs

F( ,

=

) F( ,

=

) F( ,

=

) distributive lattices , = F(?) Thm[K, Marc] Thm[Chepoi, K, Marc] bipartite cellular graphs

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SLIDE 92

Let G partial cube, then G′ ⊂ G convex ⇐ ⇒ G′ restriction of G

From partial cubes to sign vectors

shortest paths between vertices of G′ stay in G′

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SLIDE 93

Let G partial cube, then G′ ⊂ G convex ⇐ ⇒ G′ restriction of G

From partial cubes to sign vectors

shortest paths between vertices of G′ stay in G′ intersection of halfspaces X(G′) containing G′

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SLIDE 94

Let G partial cube, then G′ ⊂ G convex ⇐ ⇒ G′ restriction of G

associate convex subgraph G′ with sign vector X(G′)

From partial cubes to sign vectors

shortest paths between vertices of G′ stay in G′ intersection of halfspaces X(G′) containing G′

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SLIDE 95

Let G partial cube, then G′ ⊂ G convex ⇐ ⇒ G′ restriction of G

associate convex subgraph G′ with sign vector X(G′)

From partial cubes to sign vectors

shortest paths between vertices of G′ stay in G′ intersection of halfspaces X(G′) containing G′

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SLIDE 96

Let G partial cube, then G′ ⊂ G convex ⇐ ⇒ G′ restriction of G

associate convex subgraph G′ with sign vector X(G′)

++– – + ( )

From partial cubes to sign vectors

shortest paths between vertices of G′ stay in G′ intersection of halfspaces X(G′) containing G′

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SLIDE 97

Let G partial cube, then G′ ⊂ G convex ⇐ ⇒ G′ restriction of G

associate convex subgraph G′ with sign vector X(G′)

++– – + ( ) +– – – 0 ( )

From partial cubes to sign vectors

shortest paths between vertices of G′ stay in G′ intersection of halfspaces X(G′) containing G′

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SLIDE 98

Let G partial cube, then G′ ⊂ G convex ⇐ ⇒ G′ restriction of G

associate convex subgraph G′ with sign vector X(G′)

++– – + ( ) +– – – 0 ( ) 0 0 – –0 ( )

From partial cubes to sign vectors

shortest paths between vertices of G′ stay in G′ intersection of halfspaces X(G′) containing G′

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SLIDE 99

Let G partial cube, then G′ ⊂ G convex ⇐ ⇒ G′ restriction of G

associate convex subgraph G′ with sign vector X(G′)

++– – + ( ) +– – – 0 ( ) 0 0 – –0 ( )

From partial cubes to sign vectors

L = {X(G′) | G′ ⊆ G convex } ⊆ {0, ±}C shortest paths between vertices of G′ stay in G′ intersection of halfspaces X(G′) containing G′

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SLIDE 100

Let G partial cube, then G′ ⊂ G convex ⇐ ⇒ G′ restriction of G

associate convex subgraph G′ with sign vector X(G′)

++– – + ( ) +– – – 0 ( ) 0 0 – –0 ( )

From partial cubes to sign vectors

L = {X(G′) | G′ ⊆ G convex } ⊆ {0, ±}C tope graph GL = L ∩ {±1}C ⊆ QC of L is G shortest paths between vertices of G′ stay in G′ intersection of halfspaces X(G′) containing G′

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SLIDE 101

v

G′

v′

gate of v in G′

G′ ⊆ G gated if ∀v ∈ G ∃v′ ∈ G′ s.th ∀w ∈ G′ there is a shortest (v, w)-path through v′

Gated subgraphs

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SLIDE 102

v

G′

v′

gate of v in G′ convex

G′ ⊆ G gated if ∀v ∈ G ∃v′ ∈ G′ s.th ∀w ∈ G′ there is a shortest (v, w)-path through v′

Gated subgraphs

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SLIDE 103

v

G′

no colors from G′ v′

gate of v in G′ convex

G′ ⊆ G gated if ∀v ∈ G ∃v′ ∈ G′ s.th ∀w ∈ G′ there is a shortest (v, w)-path through v′

Gated subgraphs

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SLIDE 104

v

G′

no colors from G′ v′

gate of v in G′ convex

G′ ⊆ G gated if ∀v ∈ G ∃v′ ∈ G′ s.th ∀w ∈ G′ there is a shortest (v, w)-path through v′ X(G′) ◦ X(v) = X(v′)

Gated subgraphs

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SLIDE 105

v

G′

no colors from G′ v′

gate of v in G′ convex

G′ ⊆ G gated if ∀v ∈ G ∃v′ ∈ G′ s.th ∀w ∈ G′ there is a shortest (v, w)-path through v′ X(G′) ◦ X(v) = X(v′)

Gated subgraphs

L = {X(G′) | G′ ⊆ G gated } ⊆ {0, ±}C

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SLIDE 106

v

G′

no colors from G′ v′

gate of v in G′ convex

G′ ⊆ G gated if ∀v ∈ G ∃v′ ∈ G′ s.th ∀w ∈ G′ there is a shortest (v, w)-path through v′ X(G′) ◦ X(v) = X(v′)

Gated subgraphs

L = {X(G′) | G′ ⊆ G gated } ⊆ {0, ±}C L has L ◦ L ⊆ L while GL = G

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SLIDE 107

G′ antipodal if ∀v ∈ G′ ∃v′ ∈ G′ s. th. ∀w ∈ G′ there is a shortest (v, v′)-path through w ((antipodal ⇒ convex) v′ v

Antipodal gated subgraphs

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SLIDE 108

G′ antipodal if ∀v ∈ G′ ∃v′ ∈ G′ s. th. ∀w ∈ G′ there is a shortest (v, v′)-path through w ((antipodal ⇒ convex) ⇔ X(v′) = X(G′) ◦ −X(v) v′ v

Antipodal gated subgraphs

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SLIDE 109

G′ antipodal if ∀v ∈ G′ ∃v′ ∈ G′ s. th. ∀w ∈ G′ there is a shortest (v, v′)-path through w ((antipodal ⇒ convex) ⇔ X(v′) = X(G′) ◦ −X(v) G′ ◦ −v v

Antipodal gated subgraphs

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SLIDE 110

G′ antipodal if ∀v ∈ G′ ∃v′ ∈ G′ s. th. ∀w ∈ G′ there is a shortest (v, v′)-path through w ((antipodal ⇒ convex) ⇔ X(v′) = X(G′) ◦ −X(v) antipodal and gated

Antipodal gated subgraphs

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SLIDE 111

G′ antipodal if ∀v ∈ G′ ∃v′ ∈ G′ s. th. ∀w ∈ G′ there is a shortest (v, v′)-path through w ((antipodal ⇒ convex) gated and not antipodal ⇔ X(v′) = X(G′) ◦ −X(v) antipodal and gated

Antipodal gated subgraphs

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SLIDE 112

G′ antipodal if ∀v ∈ G′ ∃v′ ∈ G′ s. th. ∀w ∈ G′ there is a shortest (v, v′)-path through w ((antipodal ⇒ convex) gated and not antipodal convex, not gated nor antipodal ⇔ X(v′) = X(G′) ◦ −X(v) antipodal and gated

Antipodal gated subgraphs

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SLIDE 113

G′ antipodal if ∀v ∈ G′ ∃v′ ∈ G′ s. th. ∀w ∈ G′ there is a shortest (v, v′)-path through w ((antipodal ⇒ convex) gated and not antipodal convex, not gated nor antipodal ⇔ X(v′) = X(G′) ◦ −X(v) antipodal and gated antipodal and not gated

Antipodal gated subgraphs

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SLIDE 114

G′ antipodal if ∀v ∈ G′ ∃v′ ∈ G′ s. th. ∀w ∈ G′ there is a shortest (v, v′)-path through w ((antipodal ⇒ convex) gated and not antipodal convex, not gated nor antipodal ⇔ X(v′) = X(G′) ◦ −X(v) antipodal and gated antipodal and not gated

Antipodal gated subgraphs

G′ antipodal and gated ⇔ every v has gate with antipode in G′ ⇔ X(G′) ◦ −X(v) ∈ {X(v′) | v′ ∈ V }

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SLIDE 115

G′ antipodal if ∀v ∈ G′ ∃v′ ∈ G′ s. th. ∀w ∈ G′ there is a shortest (v, v′)-path through w ((antipodal ⇒ convex) gated and not antipodal convex, not gated nor antipodal ⇔ X(v′) = X(G′) ◦ −X(v) antipodal and gated antipodal and not gated

Antipodal gated subgraphs

G′ antipodal and gated ⇔ every v has gate with antipode in G′ ⇔ X(G′) ◦ −X(v) ∈ {X(v′) | v′ ∈ V }

L = {X(G′) | G′ ⊆ G antipodal and gated } ⊆ {0, ±}C (FS) L ◦ −L ⊆ L

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SLIDE 116

G′ antipodal if ∀v ∈ G′ ∃v′ ∈ G′ s. th. ∀w ∈ G′ there is a shortest (v, v′)-path through w ((antipodal ⇒ convex) gated and not antipodal convex, not gated nor antipodal ⇔ X(v′) = X(G′) ◦ −X(v) antipodal and gated antipodal and not gated

Antipodal gated subgraphs

G′ antipodal and gated ⇔ every v has gate with antipode in G′ ⇔ X(G′) ◦ −X(v) ∈ {X(v′) | v′ ∈ V }

L = {X(G′) | G′ ⊆ G antipodal and gated } ⊆ {0, ±}C (FS) L ◦ −L ⊆ L

G tope graph of COM = ⇒ antipodal subgraphs gated

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SLIDE 117

Antipodal gated partial cubes and Q−

AG= {G partial cube| all antipodal subgraphs gated}

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SLIDE 118

Antipodal gated partial cubes and Q−

AG= {G partial cube| all antipodal subgraphs gated}

/ ∈ AG

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SLIDE 119

Antipodal gated partial cubes and Q−

AG= {G partial cube| all antipodal subgraphs gated}

/ ∈ AG contract

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SLIDE 120

Antipodal gated partial cubes and Q−

AG= {G partial cube| all antipodal subgraphs gated}

/ ∈ AG contract restrict

,

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SLIDE 121

Antipodal gated partial cubes and Q−

AG= {G partial cube| all antipodal subgraphs gated}

/ ∈ AG contract restrict restrict

, ,

∈ AG

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SLIDE 122

Antipodal gated partial cubes and Q−

AG= {G partial cube| all antipodal subgraphs gated}

all these are minor-minimally non AG

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SLIDE 123

Antipodal gated partial cubes and Q−

AG= {G partial cube| all antipodal subgraphs gated}

all these are minor-minimally non AG but more generally: Qd

slide-124
SLIDE 124

Antipodal gated partial cubes and Q−

AG= {G partial cube| all antipodal subgraphs gated}

all these are minor-minimally non AG but more generally: Qd Qd

slide-125
SLIDE 125

Antipodal gated partial cubes and Q−

AG= {G partial cube| all antipodal subgraphs gated}

all these are minor-minimally non AG but more generally: Qd Qd Qd Qd

slide-126
SLIDE 126

Antipodal gated partial cubes and Q−

AG= {G partial cube| all antipodal subgraphs gated}

all these are minor-minimally non AG but more generally: Qd Qd Qd Qd Qd Qd Qd Qd

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SLIDE 127

Antipodal gated partial cubes and Q−

AG= {G partial cube| all antipodal subgraphs gated}

all these are minor-minimally non AG but more generally: Qd Qd Qd Qd Qd Qd Qd Qd

Q−

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SLIDE 128

Antipodal gated partial cubes and Q−

AG= {G partial cube| all antipodal subgraphs gated}

all these are minor-minimally non AG but more generally: Qd Qd Qd Qd Qd Qd Qd Qd

Q− Lemma: AG is minor-closed

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SLIDE 129

Antipodal gated partial cubes and Q−

AG= {G partial cube| all antipodal subgraphs gated}

all these are minor-minimally non AG but more generally: Qd Qd Qd Qd Qd Qd Qd Qd

Q− Lemma: AG is minor-closed = ⇒ AG ⊆ F(Q−)

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SLIDE 130

THM[K, Marc ’17]: for a partial cube G the following are equivalent:

  • G is tope graph of a COM
  • all antipodal subgraphs of G are gated
  • G has no partial cube minor from Q−

Corollaries:

  • characterization, recognition for oriented

matroids and affine oriented matroids

  • polytime recognition

Characterization

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SLIDE 131

THM[K, Marc 17]: G tope graph of COM iff G partial cube such that all antipodal subgraphs gated. COR: G tope graph of OM iff G antipodal partial cube such that all antipodal subgraphs gated. COR: G tope graph of AOM iff G affine partial cube such that all antipodal and conformal subgraphs gated.

A common generalization

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SLIDE 132
  • check if partial cube

O(n2)

  • find antipodal subgraphs

O(n2) shortest path intervals – check if antipodal

  • for each check if gated

do some distances

naive polytime alogrithm THM[K, Marc 17]: G tope graph of COM iff G partial cube such that all antipodal subgraphs gated.

Recognition

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SLIDE 133

Further questions

Observation: tope graphs of realizable COMs are convex subgraphs of tope graphs of realizable OMs.

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SLIDE 134

Further questions

Observation: tope graphs of realizable COMs are convex subgraphs of tope graphs of realizable OMs.

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SLIDE 135

Further questions

Observation: tope graphs of realizable COMs are convex subgraphs of tope graphs of realizable OMs.

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SLIDE 136

Further questions

Observation: tope graphs of realizable COMs are convex subgraphs of tope graphs of realizable OMs.

Conjecture [Bandelt, Chepoi, K ’15]: every GCOM is convex subgraph of GOM.

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SLIDE 137

Further questions

Observation: tope graphs of realizable COMs are convex subgraphs of tope graphs of realizable OMs.

Conjecture [Bandelt, Chepoi, K ’15]: every GCOM is convex subgraph of GOM.

would yield a Topological Representation Theorem with pseudohyperplanes and pseudohalfspaces for COMs

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SLIDE 138

Further questions

Observation: tope graphs of realizable COMs are convex subgraphs of tope graphs of realizable OMs.

Conjecture [Bandelt, Chepoi, K ’15]: every GCOM is convex subgraph of GOM.

would yield a Topological Representation Theorem with pseudohyperplanes and pseudohalfspaces for COMs

Find excluded minors for:

  • planar partial cubes
  • realizable COMs
  • flip graphs of acyclic orientations of

mixed graphs

slide-139
SLIDE 139

Further questions

Observation: tope graphs of realizable COMs are convex subgraphs of tope graphs of realizable OMs.

Conjecture [Bandelt, Chepoi, K ’15]: every GCOM is convex subgraph of GOM.

would yield a Topological Representation Theorem with pseudohyperplanes and pseudohalfspaces for COMs

Find excluded minors for:

  • planar partial cubes
  • realizable COMs
  • flip graphs of acyclic orientations of

mixed graphs

Merci!