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

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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

  1. From graphic to realizable 5 digraph D = ( V, E ) − a − 1 1 minimal edge cut X 6 − b + 1 incidence matrix I ∈ {± 1 , 0 } V × E 2 c + . . . a b c d e 3 7 + +1 ·   − 1 0 0 0 0 . . . d e + 8 +1 · − 0 + 0 0 . . . 2 4     0 0 0 + 0 . . . +1 · 3     a +1 · 0 0 0 0 + . . . 4     + 0 0 0 0 . . . − 1 · 5   c   − 1 · 0 + 0 0 0 . . . 6     − 1 · 7 0 0 − − 0 . . .   b   − 1 · − 0 0 0 0 . . . 8 d   . . . . . .  ...  . . . . . . . . . . . . e − 2 − 2 +2 +2 +2 Σ = ( 0 . . . 0 ) � − 0 . . . 0 � = X ∈ C ∗ − + + + sgn( Σ) = C ∗ := support-minimal sign vectors of elements of row-space of I

  2. From graphic to realizable 5 digraph D = ( V, E ) − a − 1 1 minimal edge cut X 6 − b + 1 incidence matrix I ∈ {± 1 , 0 } V × E 2 c + . . . a b c d e 3 7 + +1 ·   − 1 0 0 0 0 . . . d e + 8 +1 · − 0 + 0 0 . . . 2 4     0 0 0 + 0 . . . +1 · 3     a +1 · 0 0 0 0 + . . . 4     + 0 0 0 0 . . . − 1 · 5   c   − 1 · 0 + 0 0 0 . . . 6   X   − 1 · 7 0 0 − − 0 . . .   b   − 1 · − 0 0 0 0 . . . 8 d   . . . . . .  ...  . . . . . . . . . . . . e − 2 − 2 +2 +2 +2 Σ = ( 0 . . . 0 ) � − 0 . . . 0 � = X ∈ C ∗ − + + + sgn( Σ) = C ∗ := support-minimal sign vectors of elements of row-space of I

  3. From graphic to realizable 5 digraph D = ( V, E ) − a − 1 1 minimal edge cut X 6 − b + 1 incidence matrix I ∈ {± 1 , 0 } V × E 2 c + . . . a b c d e 3 7 + +1 ·   − 1 0 0 0 0 . . . d e + 8 +1 · − 0 + 0 0 . . . 2 4     0 0 0 + 0 . . . +1 · 3     a +1 · 0 0 0 0 + . . . 4     + 0 0 0 0 . . . − 1 · 5   c   − 1 · 0 + 0 0 0 . . . 6   X   − 1 · 7 0 0 − − 0 . . .   b   − 1 · − 0 0 0 0 . . . 8 d   . . . . . .  ...  . . . . . . . . . . . . e − 2 − 2 +2 +2 +2 Σ = ( 0 . . . 0 ) � − 0 . . . 0 � = X ∈ C ∗ − + + + sgn( Σ) = C ∗ := support-minimal sign vectors sign vectors of min-dimensional = of elements of row-space of I cells of hyperplane arrangement

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

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

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

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

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

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

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

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

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

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

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

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

  16. Acyclic orientations directed 5 digraph D = ( V, E ) − a − 1 1 minimal edge cut X 6 − b + 1 incidence matrix I ∈ {± 1 , 0 } V × E 2 c + . . . a b c d e 3 7 + +1 ·   − + 1 0 0 0 0 . . . d e + 8 +1 · − 0 + + 0 0 . . . 2 4     0 0 0 + 0 . . . +1 · 3     a +1 · 0 0 0 0 + . . . 4     − + 0 0 0 0 . . . − 1 · 5 reorientation   c   − 1 · − 0 + 0 0 0 . . . 6   X   − 1 · 7 0 0 − − 0 . . .   b   − 1 · − 0 0 0 0 . . . 8 d   . . . . . .  ...  . . . . . . . . . . . . e � − 0 . . . 0 � + − + + + + = X sgn( Σ) = every edge in a directed cut ⇐ ⇒ acyclic orientation

  17. Acyclic orientations directed 5 digraph D = ( V, E ) − a − 1 1 minimal edge cut X 6 − b + 1 incidence matrix I ∈ {± 1 , 0 } V × E 2 c + . . . a b c d e 3 7 + +1 ·   + − 1 0 0 0 0 . . . d e + 8 +1 · − 0 + + 0 0 . . . 2 4     0 0 0 + 0 . . . +1 · 3     a +1 · 0 0 0 0 + . . . 4     − + 0 0 0 0 . . . − 1 · 5 reorientation   c   − 1 · − 0 + 0 0 0 . . . 6   X   − 1 · 7 0 0 − − 0 . . .   b   − 1 · − 0 0 0 0 . . . 8 d   . . . . . .  ...  . . . . . . . . . . . . e � − 0 . . . 0 � + − + + + + = X sgn( Σ) = every edge in a directed cut max-dimensional cells ∼ ⇐ ⇒ acyclic orientation = acyclic orientations

  18. Acyclic orientations directed 5 digraph D = ( V, E ) − a − 1 1 minimal edge cut X 6 − b + 1 incidence matrix I ∈ {± 1 , 0 } V × E 2 c + . . . a b c d e 3 7 + +1 ·   + − 1 0 0 0 0 . . . d e + 8 +1 · − 0 + + 0 0 . . . 2 4     0 0 0 + 0 . . . +1 · 3     a +1 · 0 0 0 0 + . . . 4     − + 0 0 0 0 . . . − 1 · 5 reorientation   c   − 1 · − 0 + 0 0 0 . . . 6   X   − 1 · 7 0 0 − − 0 . . .   b   − 1 · − 0 0 0 0 . . . 8 d   . . . . . .  ...  . . . . . . . . . . . . e � − 0 . . . 0 � + − + + + + = X sgn( Σ) = every edge in a directed cut max-dimensional cells ∼ ⇐ ⇒ acyclic orientation = acyclic orientations

  19. Acyclic orientations directed 5 digraph D = ( V, E ) − a − 1 1 minimal edge cut X 6 − b + 1 incidence matrix I ∈ {± 1 , 0 } V × E 2 c + . . . a b c d e 3 7 + +1 ·   + − 1 0 0 0 0 . . . d e + 8 +1 · − 0 + + 0 0 . . . 2 4     0 0 0 + 0 . . . +1 · 3     a +1 · 0 0 0 0 + . . . 4     − + 0 0 0 0 . . . − 1 · 5 reorientation   c   − 1 · − 0 + 0 0 0 . . . 6   X   − 1 · 7 0 0 − − 0 . . .   b   − 1 · − 0 0 0 0 . . . 8 d   . . . . . .  ...  . . . . . . . . . . . . e � − 0 . . . 0 � + − + + + + = X sgn( Σ) = every edge in a directed cut max-dimensional cells ∼ ⇐ ⇒ acyclic orientation = acyclic orientations

  20. Acyclic orientations directed 5 digraph D = ( V, E ) − a − 1 1 minimal edge cut X 6 − b + 1 incidence matrix I ∈ {± 1 , 0 } V × E 2 c + . . . a b c d e 3 7 + +1 ·   − + 1 0 0 0 0 . . . d e + 8 +1 · − 0 + + 0 0 . . . 2 4     0 0 0 + 0 . . . +1 · 3     a +1 · 0 0 0 0 + . . . 4     − + 0 0 0 0 . . . − 1 · 5 reorientation   c   − 1 · − 0 + 0 0 0 . . . 6   X   − 1 · 7 0 0 − − 0 . . .   b   − 1 · − 0 0 0 0 . . . 8 d   . . . . . .  ...  . . . . . . . . . . . . e � − 0 . . . 0 � + − + + + + = X sgn( Σ) = flip-graph on acyclic orientations ∼ = region every edge in a directed cut max-dimensional cells ∼ ⇐ ⇒ acyclic orientation graph of arrangement = acyclic orientations

  21. Realizable oriented matroids C ∗ := support-minimal sign vectors of elements of R -vector space = sign vectors of min-dimensional cells of (central) hyperplane arrangement

  22. Realizable oriented matroids C ∗ := support-minimal sign vectors of elements of R -vector space = sign vectors of min-dimensional cells of (central) hyperplane arrangement

  23. Realizable oriented matroids C ∗ := support-minimal sign vectors of elements of R -vector space = sign vectors of min-dimensional cells of (central) hyperplane arrangement

  24. Realizable oriented matroids C ∗ := support-minimal sign vectors sign vectors of min-dimensional cells of elements of R -vector space = of (central) hyperplane arrangement sign vectors of 0 -dimensional cells of = arrangement of great cycles on sphere

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

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

  27. Oriented matroids Thm[Folkman, Lawrence ’78] correspondence of pseudo-sphere arrangements and oriented matroids . 0 + − 0

  28. Oriented matroids Thm[Folkman, Lawrence ’78] correspondence of pseudo-sphere arrangements and oriented matroids . + − − + 0 + − 0

  29. Oriented matroids Thm[Folkman, Lawrence ’78] correspondence of pseudo-sphere arrangements and oriented +0 − + matroids . + − − + 0 + − 0

  30. Oriented matroids Thm[Folkman, Lawrence ’78] correspondence of pseudo-sphere arrangements and oriented +0 − + matroids . + − − + 0 + − 0 ◦ Covector axioms: ( E, L ) OM iff (Z) 0 ∈ L (FS) L ◦ −L ⊆ L (SE) ∀ X, Y ∈ L and e ∈ S ( X, Y ) ∃ Z ∈ L : Z e = 0 and Z f = X f ◦ Y f for f / ∈ S ( X, Y ) .

  31. Oriented matroids Thm[Folkman, Lawrence ’78] correspondence of pseudo-sphere arrangements and oriented matroids . + − − + topes T of M = maximal cells= ◦ Covector axioms: ( E, L ) OM iff max. covectors (Z) 0 ∈ L (FS) L ◦ −L ⊆ L (SE) ∀ X, Y ∈ L and e ∈ S ( X, Y ) ∃ Z ∈ L : Z e = 0 and Z f = X f ◦ Y f for f / ∈ S ( X, Y ) .

  32. Oriented matroids Thm[Folkman, Lawrence ’78] correspondence of pseudo-sphere arrangements and oriented matroids . + − − + topes T of M = maximal cells= ◦ Covector axioms: ( E, L ) OM iff max. covectors (Z) 0 ∈ L (FS) L ◦ −L ⊆ L (SE) ∀ X, Y ∈ L and e ∈ S ( X, Y ) ∃ Z ∈ L : Z e = 0 and Z f = X f ◦ Y f for f / ∈ S ( X, Y ) .

  33. Oriented matroids Thm[Folkman, Lawrence ’78] correspondence of pseudo-sphere arrangements and oriented matroids . + − − + topes T of M = maximal cells= ◦ Covector axioms: ( E, L ) OM iff max. covectors (Z) 0 ∈ L (FS) L ◦ −L ⊆ L tope graph G L (SE) ∀ X, Y ∈ L and e ∈ S ( X, Y ) ∃ Z ∈ L : =incidence graph= Z e = 0 and Z f = X f ◦ Y f for f / ∈ S ( X, Y ) . induced graph in Q E

  34. Affine oriented matroids Thm[Karlander ’92] correspondence between affine arrangements of pseudospheres and affine oriented matroids . + − − + 0 + − 0

  35. Affine oriented matroids Thm[Karlander ’92] correspondence between affine arrangements of pseudospheres and affine oriented matroids . + − − + 0 + − 0

  36. 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 : Z e = 0 and Z f = X f ◦ Y f for f / ∈ S ( X, Y ) .

  37. Affine oriented matroids Thm[Karlander ’92] correspondence between affine arrangements of pseudospheres and affine oriented matroids . + − − + 0 + − 0 topes T of M = maximal cells= ◦ Covector axioms: ( E, L ) affine oriented matroid: max. covectors (A) something lengthy tope graph G L (FS) L ◦ −L ⊆ L =incidence graph= (SE) ∀ X, Y ∈ L and e ∈ S ( X, Y ) ∃ Z ∈ L : induced graph in Q E Z e = 0 and Z f = X f ◦ Y f for f / ∈ S ( X, Y ) .

  38. Affine oriented matroids Thm[Karlander ’92] e correspondence between affine arrangements of pseudospheres and affine oriented matroids . + − − + digraph example: 0 + − 0 topes T ∼ = acyclic orientations with edge e ’s orientation fixed topes T of M = maximal cells= ◦ Covector axioms: ( E, L ) affine oriented matroid: max. covectors (A) something lengthy tope graph G L (FS) L ◦ −L ⊆ L =incidence graph= (SE) ∀ X, Y ∈ L and e ∈ S ( X, Y ) ∃ Z ∈ L : induced graph in Q E Z e = 0 and Z f = X f ◦ Y f for f / ∈ S ( X, Y ) .

  39. Affine oriented matroids Thm[Karlander ’92] e correspondence between affine arrangements of pseudospheres and affine oriented matroids . + − − + digraph example: 0 + − 0 topes T ∼ = acyclic orientations with edge e ’s orientation fixed Bandelt, Chepoi, K ’15: why not fix more? topes T of M = maximal cells= ◦ Covector axioms: ( E, L ) affine oriented matroid: max. covectors (A) something lengthy tope graph G L (FS) L ◦ −L ⊆ L =incidence graph= (SE) ∀ X, Y ∈ L and e ∈ S ( X, Y ) ∃ Z ∈ L : induced graph in Q E Z e = 0 and Z f = X f ◦ Y f for f / ∈ S ( X, Y ) .

  40. Complexes of oriented matroids + − − + 0 + − 0 Bandelt, Chepoi, K ’15: why not fix more? topes T of M = maximal cells= max. covectors ◦ Covector axioms: ( E, L ) COM iff (FS) L ◦ −L ⊆ L tope graph G L =incidence graph= (SE) ∀ X, Y ∈ L and e ∈ S ( X, Y ) ∃ Z ∈ L : Z e = 0 and Z f = X f ◦ Y f for f / ∈ S ( X, Y ) . induced graph in Q E

  41. Complexes of oriented matroids Def[Bandelt, Chepoi, K ’15] realizable COM = sign systems from arrangement of open halfspaces and hyperplanes. topes T of M = maximal cells= max. covectors ◦ Covector axioms: ( E, L ) COM iff (FS) L ◦ −L ⊆ L tope graph G L =incidence graph= (SE) ∀ X, Y ∈ L and e ∈ S ( X, Y ) ∃ Z ∈ L : Z e = 0 and Z f = X f ◦ Y f for f / ∈ S ( X, Y ) . induced graph in Q E

  42. Complexes of oriented matroids Def[Bandelt, Chepoi, K ’15] realizable COM = sign systems from arrangement of open halfspaces and hyperplanes.       − 0 + + + +        ◦ ( − (FS)  ) =       − − +     + + + topes T of M = maximal cells= max. covectors ◦ Covector axioms: ( E, L ) COM iff (FS) L ◦ −L ⊆ L tope graph G L =incidence graph= (SE) ∀ X, Y ∈ L and e ∈ S ( X, Y ) ∃ Z ∈ L : Z e = 0 and Z f = X f ◦ Y f for f / ∈ S ( X, Y ) . induced graph in Q E

  43. Complexes of oriented matroids Def[Bandelt, Chepoi, K ’15] realizable COM = sign systems from arrangement of open halfspaces and hyperplanes.       − − 0 + − +        ◦ (FS)  =       − − −     − + + topes T of M = maximal cells= max. covectors ◦ Covector axioms: ( E, L ) COM iff (FS) L ◦ −L ⊆ L tope graph G L =incidence graph= (SE) ∀ X, Y ∈ L and e ∈ S ( X, Y ) ∃ Z ∈ L : Z e = 0 and Z f = X f ◦ Y f for f / ∈ S ( X, Y ) . induced graph in Q E

  44. Complexes of oriented matroids Def[Bandelt, Chepoi, K ’15] realizable COM = sign systems from arrangement of open X halfspaces and hyperplanes.       − − 0 + − +        ◦ (FS)  =       − − − Y     − + + topes T of M = maximal cells= max. covectors ◦ Covector axioms: ( E, L ) COM iff (FS) L ◦ −L ⊆ L tope graph G L =incidence graph= (SE) ∀ X, Y ∈ L and e ∈ S ( X, Y ) ∃ Z ∈ L : Z e = 0 and Z f = X f ◦ Y f for f / ∈ S ( X, Y ) . induced graph in Q E

  45. Complexes of oriented matroids Def[Bandelt, Chepoi, K ’15] realizable COM = sign systems from arrangement of open X halfspaces and hyperplanes.       − − 0 + − +        ◦ (FS)  =       − − − Y     − + + topes T of M = maximal cells= max. covectors ◦ Covector axioms: ( E, L ) COM iff (FS) L ◦ −L ⊆ L tope graph G L =incidence graph= (SE) ∀ X, Y ∈ L and e ∈ S ( X, Y ) ∃ Z ∈ L : Z e = 0 and Z f = X f ◦ Y f for f / ∈ S ( X, Y ) . induced graph in Q E

  46. Complexes of oriented matroids Def[Bandelt, Chepoi, K ’15] realizable COM = sign systems X ◦ − Y from arrangement of open X halfspaces and hyperplanes.       − − 0 + − +        ◦ (FS)  =       − − − Y     − + + topes T of M = maximal cells= max. covectors ◦ Covector axioms: ( E, L ) COM iff (FS) L ◦ −L ⊆ L tope graph G L =incidence graph= (SE) ∀ X, Y ∈ L and e ∈ S ( X, Y ) ∃ Z ∈ L : Z e = 0 and Z f = X f ◦ Y f for f / ∈ S ( X, Y ) . induced graph in Q E

  47. Complexes of oriented matroids Def[Bandelt, Chepoi, K ’15] realizable COM = sign systems from arrangement of open halfspaces and hyperplanes.     0 − + − topes T of M =      , (SE)     − − maximal cells=    − + max. covectors ◦ Covector axioms: ( E, L ) COM iff (FS) L ◦ −L ⊆ L tope graph G L =incidence graph= (SE) ∀ X, Y ∈ L and e ∈ S ( X, Y ) ∃ Z ∈ L : Z e = 0 and Z f = X f ◦ Y f for f / ∈ S ( X, Y ) . induced graph in Q E

  48. Complexes of oriented matroids Def[Bandelt, Chepoi, K ’15] realizable COM = sign systems from arrangement of open halfspaces and hyperplanes.     0 − e + − topes T of M =      , (SE)     − − maximal cells=    − + max. covectors ◦ Covector axioms: ( E, L ) COM iff (FS) L ◦ −L ⊆ L tope graph G L =incidence graph= (SE) ∀ X, Y ∈ L and e ∈ S ( X, Y ) ∃ Z ∈ L : Z e = 0 and Z f = X f ◦ Y f for f / ∈ S ( X, Y ) . induced graph in Q E

  49. Complexes of oriented matroids Def[Bandelt, Chepoi, K ’15] realizable COM = sign systems from arrangement of open halfspaces and hyperplanes.     0 −   − e + − 0 topes T of M =        , (SE) �     − −   − maximal cells=      − + ? max. covectors ◦ Covector axioms: ( E, L ) COM iff (FS) L ◦ −L ⊆ L tope graph G L =incidence graph= (SE) ∀ X, Y ∈ L and e ∈ S ( X, Y ) ∃ Z ∈ L : Z e = 0 and Z f = X f ◦ Y f for f / ∈ S ( X, Y ) . induced graph in Q E

  50. Complexes of oriented matroids Def[Bandelt, Chepoi, K ’15] realizable COM = sign systems from arrangement of open X halfspaces and hyperplanes. e Y     0 −   − e + − 0 topes T of M =        , (SE) �     − −   − maximal cells=      − + ? max. covectors ◦ Covector axioms: ( E, L ) COM iff (FS) L ◦ −L ⊆ L tope graph G L =incidence graph= (SE) ∀ X, Y ∈ L and e ∈ S ( X, Y ) ∃ Z ∈ L : Z e = 0 and Z f = X f ◦ Y f for f / ∈ S ( X, Y ) . induced graph in Q E

  51. Complexes of oriented matroids Def[Bandelt, Chepoi, K ’15] realizable COM = sign systems from arrangement of open X halfspaces and hyperplanes. Z e Y     0 −   − e + − 0 topes T of M =        , (SE) �     − −   − maximal cells=      − + ? max. covectors ◦ Covector axioms: ( E, L ) COM iff (FS) L ◦ −L ⊆ L tope graph G L =incidence graph= (SE) ∀ X, Y ∈ L and e ∈ S ( X, Y ) ∃ Z ∈ L : Z e = 0 and Z f = X f ◦ Y f for f / ∈ S ( X, Y ) . induced graph in Q E

  52. Complexes of oriented matroids e 2 Def[Bandelt, Chepoi, K ’15] e 3 realizable COM = sign systems e 1 from arrangement of open e 4 halfspaces and hyperplanes. e 9 e 5 digraph example: topes T ∼ = acyclic orientations e 6 e 8 with edges E ’s orientation fixed e 7 topes T of M = maximal cells= max. covectors ◦ Covector axioms: ( E, L ) COM iff (FS) L ◦ −L ⊆ L tope graph G L =incidence graph= (SE) ∀ X, Y ∈ L and e ∈ S ( X, Y ) ∃ Z ∈ L : Z e = 0 and Z f = X f ◦ Y f for f / ∈ S ( X, Y ) . induced graph in Q E

  53. Complexes of oriented matroids Def[Bandelt, Chepoi, K ’15] realizable COM = sign systems from arrangement of open halfspaces and hyperplanes. topes T ∼ = acyclic orientations of a mixed graph topes T of M = maximal cells= max. covectors ◦ Covector axioms: ( E, L ) COM iff (FS) L ◦ −L ⊆ L tope graph G L =incidence graph= (SE) ∀ X, Y ∈ L and e ∈ S ( X, Y ) ∃ Z ∈ L : Z e = 0 and Z f = X f ◦ Y f for f / ∈ S ( X, Y ) . induced graph in Q E

  54. A common generalization ◦ Covector axioms: ( E, L ) COM iff (FS) L ◦ −L ⊆ L (SE) ∀ X, Y ∈ L and e ∈ S ( X, Y ) ∃ Z ∈ L : Z e = 0 and Z f = X f ◦ Y f for f / ∈ S ( X, Y ) . ◦ Covector axioms: ( E, L ) oriented matroid: (Z) ∅ ∈ L (FS) L ◦ −L ⊆ L (SE) ∀ X, Y ∈ L and e ∈ S ( X, Y ) ∃ Z ∈ L : Z e = 0 and Z f = X f ◦ Y f 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 : Z e = 0 and Z f = X f ◦ Y f for f / ∈ S ( X, Y ) .

  55. A common generalization ◦ Covector axioms: ( E, L ) COM iff (FS) L ◦ −L ⊆ L (SE) ∀ X, Y ∈ L and e ∈ S ( X, Y ) ∃ Z ∈ L : Z e = 0 and Z f = X f ◦ Y f for f / ∈ S ( X, Y ) . ◦ Covector axioms: ( E, L ) oriented matroid: L e (Z) ∅ ∈ L n i m r (FS) L ◦ −L ⊆ L e t e d (SE) ∀ X, Y ∈ L and e ∈ S ( X, Y ) ∃ Z ∈ L : d n a Z e = 0 and Z f = X f ◦ Y f for f / ∈ S ( X, Y ) . s e b u c ◦ Covector axioms: ( E, L ) affine oriented matroid: l a i t r a p (A) something lengthy s h p (FS) L ◦ −L ⊆ L a r g e (SE) ∀ X, Y ∈ L and e ∈ S ( X, Y ) ∃ Z ∈ L : p o t Z e = 0 and Z f = X f ◦ Y f for f / ∈ S ( X, Y ) .

  56. Partial cubes and partial cube minors G partial cube : ⇔ G isometric subgraph of hypercube G ⊆ Q n such that d G ( v, w ) = d Q n ( v, w ) ∀ v, w ∈ G

  57. Partial cubes and partial cube minors G partial cube : ⇔ G isometric subgraph of hypercube G ⊆ Q n such that d G ( v, w ) = d Q n ( v, w ) ∀ v, w ∈ G tope graph of realizable COM (arrangement of half and hyperplanes)

  58. Partial cubes and partial cube minors G partial cube : ⇔ G isometric subgraph of hypercube G ⊆ Q n such that d G ( v, w ) = d Q n ( v, w ) ∀ v, w ∈ G tope graph of realizable COM (arrangement of half and hyperplanes)

  59. Partial cubes and partial cube minors G partial cube : ⇔ G isometric subgraph of hypercube G ⊆ Q n such that d G ( v, w ) = d Q n ( v, w ) ∀ v, w ∈ G tope graph of realizable COM (arrangement of half and hyperplanes)

  60. Partial cubes and partial cube minors G partial cube : ⇔ G isometric subgraph of hypercube G ⊆ Q n such that d G ( v, w ) = d Q n ( v, w ) ∀ v, w ∈ G edges of partial cube naturally partitioned into minimal cuts C

  61. Partial cubes and partial cube minors G partial cube : ⇔ G isometric subgraph of hypercube G ⊆ Q n such that d G ( v, w ) = d Q n ( v, w ) ∀ v, w ∈ G edges of partial cube naturally partitioned into minimal cuts C � minor-relation

  62. Partial cubes and partial cube minors G partial cube : ⇔ G isometric subgraph of hypercube G ⊆ Q n such that d G ( v, w ) = d Q n ( v, w ) ∀ v, w ∈ G restriction to a side of a cut edges of partial cube naturally partitioned into minimal cuts C � minor-relation

  63. Partial cubes and partial cube minors G partial cube : ⇔ G isometric subgraph of hypercube G ⊆ Q n such that d G ( v, w ) = d Q n ( v, w ) ∀ v, w ∈ G restriction to a side of a cut contraction of a cut edges of partial cube naturally partitioned into minimal cuts C � minor-relation

  64. Partial cubes and partial cube minors G partial cube : ⇔ G isometric subgraph of hypercube G ⊆ Q n such that d G ( v, w ) = d Q n ( v, w ) ∀ v, w ∈ G restriction to a side of a cut contraction of a cut edges of partial cube naturally partitioned into minimal cuts C � minor-relation � yields new partial cube

  65. Partial cubes and partial cube minors G partial cube : ⇔ G isometric subgraph of hypercube G ⊆ Q n such that d G ( v, w ) = d Q n ( v, w ) ∀ v, w ∈ G tope graph of realizable COM (arrangement of half and hyperplanes) edges of partial cube naturally partitioned into minimal cuts C � minor-relation � yields new tope graph

  66. Partial cube minors some minor-closed classes partial cubes G COM graphs of arrangements planar of half- and hyperplanes partial cubes graphs of acyclic Hypercellular graphs ors of mixed graphs median graphs bipartite cellular graphs distributive lattices

  67. Partial cube minors some minor-closed classes partial cubes G COM graphs of arrangements planar of half- and hyperplanes partial cubes graphs of acyclic Hypercellular graphs ors of mixed graphs median graphs bipartite cellular graphs distributive lattices each has a family of excluded minors

  68. Partial cube minors some minor-closed classes partial cubes Thm[K, Marc] F ( Q − ) = G COM graphs of arrangements planar of half- and hyperplanes partial cubes graphs of acyclic Hypercellular graphs ors of mixed graphs median graphs bipartite cellular graphs distributive lattices each has a family of excluded minors

  69. Partial cube minors some minor-closed classes partial cubes Thm[K, Marc] F ( Q − ) = G COM graphs of arrangements planar of half- and hyperplanes partial cubes graphs of acyclic Hypercellular graphs ors of mixed graphs = median graphs F ( ) = bipartite cellular graphs F ( , ) = distributive Thm[Chepoi, K, Marc] F ( , ) lattices = F ( , , ) each has a family of excluded minors

  70. Partial cube minors some minor-closed classes partial cubes Thm[K, Marc] F (?) F ( Q − ) = G COM = graphs of arrangements planar of half- and hyperplanes partial cubes graphs of acyclic = F (?) Hypercellular graphs = ors of mixed graphs F (?) = median graphs F ( ) = bipartite cellular graphs F ( , ) = distributive Thm[Chepoi, K, Marc] F ( , ) lattices = F ( , , ) each has a family of excluded minors

  71. From partial cubes to sign vectors Let G partial cube, then G ′ ⊂ G convex ⇐ ⇒ G ′ restriction of G shortest paths between vertices of G ′ stay in G ′

  72. From partial cubes to sign vectors Let G partial cube, then G ′ ⊂ G convex ⇐ ⇒ G ′ restriction of G shortest paths between intersection of halfspaces vertices of G ′ stay in G ′ X ( G ′ ) containing G ′

  73. From partial cubes to sign vectors Let G partial cube, then G ′ ⊂ G convex ⇐ ⇒ G ′ restriction of G shortest paths between intersection of halfspaces vertices of G ′ stay in G ′ X ( G ′ ) containing G ′ associate convex subgraph G ′ with sign vector X ( G ′ )

  74. From partial cubes to sign vectors Let G partial cube, then G ′ ⊂ G convex ⇐ ⇒ G ′ restriction of G shortest paths between intersection of halfspaces vertices of G ′ stay in G ′ X ( G ′ ) containing G ′ associate convex subgraph G ′ with sign vector X ( G ′ )

  75. From partial cubes to sign vectors Let G partial cube, then G ′ ⊂ G convex ⇐ ⇒ G ′ restriction of G shortest paths between intersection of halfspaces vertices of G ′ stay in G ′ X ( G ′ ) containing G ′ associate convex subgraph G ′ with sign vector X ( G ′ ) ( ++– – + )

  76. From partial cubes to sign vectors Let G partial cube, then G ′ ⊂ G convex ⇐ ⇒ G ′ restriction of G shortest paths between intersection of halfspaces vertices of G ′ stay in G ′ X ( G ′ ) containing G ′ associate convex subgraph G ′ with sign vector X ( G ′ ) ( ++– – + ) ( +– – – 0 )

  77. From partial cubes to sign vectors Let G partial cube, then G ′ ⊂ G convex ⇐ ⇒ G ′ restriction of G shortest paths between intersection of halfspaces vertices of G ′ stay in G ′ X ( G ′ ) containing G ′ associate convex subgraph G ′ with sign vector X ( G ′ ) ( 0 0 – –0 ) ( ++– – + ) ( +– – – 0 )

  78. From partial cubes to sign vectors Let G partial cube, then G ′ ⊂ G convex ⇐ ⇒ G ′ restriction of G shortest paths between intersection of halfspaces vertices of G ′ stay in G ′ X ( G ′ ) containing G ′ associate convex subgraph G ′ with sign vector X ( G ′ ) ( 0 0 – –0 ) ( ++– – + ) ( +– – – 0 ) L = { X ( G ′ ) | G ′ ⊆ G convex } ⊆ { 0 , ±} C

  79. From partial cubes to sign vectors Let G partial cube, then G ′ ⊂ G convex ⇐ ⇒ G ′ restriction of G shortest paths between intersection of halfspaces vertices of G ′ stay in G ′ X ( G ′ ) containing G ′ associate convex subgraph G ′ with sign vector X ( G ′ ) ( 0 0 – –0 ) ( ++– – + ) ( +– – – 0 ) L = { X ( G ′ ) | G ′ ⊆ G convex } ⊆ { 0 , ±} C tope graph G L = L ∩ {± 1 } C ⊆ Q C of L is G

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