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

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

Graphic oriented matroids

digraph D = (V, E)

Graphic oriented matroids

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

Graphic oriented matroids

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

Graphic oriented matroids

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

Graphic oriented matroids

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

Graphic oriented matroids

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

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

Graphic oriented matroids

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

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

cocircuit X ∈ C∗

Graphic oriented matroids

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

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

cocircuit X ∈ C∗ − − + + +

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 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

From graphic to realizable

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

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

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

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·

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 ? )

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

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

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∗

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

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

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

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

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

=

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

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

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 + −

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 + − +

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 + − +

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

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

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 + − +

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

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 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 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 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

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

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

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

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

Realizable oriented matroids

C∗ := support-minimal sign vectors

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

Realizable oriented matroids

C∗ := support-minimal sign vectors

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

Realizable oriented matroids

C∗ := support-minimal sign vectors

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

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

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

Oriented matroids

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

Oriented matroids

0 + −0

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

Oriented matroids

+ − −+ 0 + −0

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

Oriented matroids

+ − −+ 0 + −0

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

Oriented matroids

+0 − +

+ − −+ 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 − +

+ − −+

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

+ − −+

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

+ − −+

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

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

+ − −+ 0 + −0

Affine oriented matroids

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

+ − −+ 0 + −0

Affine oriented matroids

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

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

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

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:

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:

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

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

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

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

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

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

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

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

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

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

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)

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)

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)

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)

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)

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

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

• 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

• 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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′

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′

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′

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′

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′

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′

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′

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′

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′

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

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

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

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

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

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

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

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

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

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

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

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

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 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 }

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′ 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

Antipodal gated partial cubes and Q−

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

Antipodal gated partial cubes and Q−

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

/ ∈ AG

Antipodal gated partial cubes and Q−

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

/ ∈ AG contract

Antipodal gated partial cubes and Q−

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

/ ∈ AG contract restrict

,

Antipodal gated partial cubes and Q−

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

/ ∈ AG contract restrict restrict

, ,

∈ AG

Antipodal gated partial cubes and Q−

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

all these are minor-minimally non AG

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

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

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

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

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−

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

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−)

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

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

• 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

Further questions

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

Further questions

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

Further questions

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

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.

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

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

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!