Dushnik-Miller dimension of stair contact complexes
Daniel Gonçalves, Lucas Isenmann
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Theorem (with Gonçalves and Pennarun)
Any triangle free planar graph is the contact graph of thick L's in the plane.
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Generalization?
What kind of graphs can be represented as �thick L's� in Rd?
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A generalization of thick L's: stairs
In all the presentation black points will be in general position: no two black
points will share a coordinate.
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A generalization of thick L's: stairs
In all the presentation black points will be in general position: no two black
points will share a coordinate.
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Consider now a stair packing and its contact complex:
What kind of simplicial complexes can be represented as contact complexes
of stair packings?
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Consider particular kind of stairs: ordered stairs
a
b
c
a < c et b < c
Note that ordered stairs can be in�nite.
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An ordered stair system is a set of points P (in Rd ) equipped with a linear
order:
a
b
c
d
e
Figure: a < b < c < d < e
We de�ne ∆(P,≤) as the contact complex of the ordered stairs.
What are their properties?
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An ordered stair system is a set of points P (in Rd ) equipped with a linear
order:
a
b
c
d
e
Figure: a < b < c < d < e
We de�ne ∆(P,≤) as the contact complex of the ordered stairs.
What are their properties?
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An ordered stair system is a set of points P (in Rd ) equipped with a linear
order:
a
b
c
d
e
Figure: a < b < c < d < e
Theorem
Any ordered stair system is a tiling (if w ∈ S(x), then for every point z
such that wi ≤ zi , z ∈⋃
u∈P S(u)).
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Given an ordered stair system, de�ne the canonical orders:
a
b
c
d
e
b e d c a
e
c
d
a
b
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a
b
c
d
e
b e d c a
e
c
d
a
b
≤1 b e d c a
≤2 e c d a b
≤3 a b c d e
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a
b
c
d
e
b e d c a
e
c
d
a
b
≤1 b e d c a
≤2 e c d a b
≤3 a b c d e
(bed) is a face of the contact complex. Note that every point is
dominating (bed) in at least one order.
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a
b
c
d
e
b e d c a
e
c
d
a
b
≤1 b e d c a
≤2 e c d a b
≤3 a b c d e
(ae) is not a face of the contact complex. Note that every point is
dominating (ae) in at least one order.
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a
b
c
d
e
b e d c a
e
c
d
a
b
≤1 b e d c a
≤2 e c d a b
≤3 a b c d e
We can check that F is a face of the contact complex i� every point is
dominating F in at least one order.
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a
b
c
d
e
b e d c a
e
c
d
a
b
R =≤1 b e d c a
≤2 e c d a b
≤3 a b c d e
In other words ∆(P,≤) = Σ(R) where Σ(R) is a simplicial complex
de�ned only on R .
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De�nition
A d -representation R is given by d linear orders on V . We de�ne the
simplicial complex Σ(R), called supremum section, as follows: F is face of
Σ(R) i� for every v ∈ V , v is dominating F in at least one order.
Theorem
Given an ordered stair system (P,≤) of Rd , we have ∆(P,≤) = Σ(R)where R is the canonical (d + 1)-representation.
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Representations are related to Dushnik-Miller dimension:
De�nition
The Dushnik-Miller dimension of a poset (≤,V ) is de�ned as the minimum
number d such that ≤ is the intersection of d linear extensions of ≤.
A poset has Dushnik-Miller dimension 1 ⇐⇒ it is a linear order.
De�nition
The Dushnik-Miller (DM for short) dimension of a simplicial complex is
de�ned as the Dushnik-Miller dimension of its inclusion poset.
A graph has DM dimension at most 1 if it is the union of single vertices.
A graph has DM dimension at most 2 if it is the union of paths.
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Representations are related to Dushnik-Miller dimension:
De�nition
The Dushnik-Miller dimension of a poset (≤,V ) is de�ned as the minimum
number d such that ≤ is the intersection of d linear extensions of ≤.
A poset has Dushnik-Miller dimension 1 ⇐⇒ it is a linear order.
De�nition
The Dushnik-Miller (DM for short) dimension of a simplicial complex is
de�ned as the Dushnik-Miller dimension of its inclusion poset.
A graph has DM dimension at most 1 if it is the union of single vertices.
A graph has DM dimension at most 2 if it is the union of paths.
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Dushnik-Miller dimension is important. For example because of
Theorem (Schnyder)
A graph G is planar i� G has DM dimension at most 3.
Theorem (Ossona de Mendez)
Any simplicial complex ∆ which has DM dimension d + 1 has a straight
line embedding in Rd .
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Theorem (Ossona de Mendez)
A simplicial complex ∆ has DM dimension at most d i� there exists a
d-representation R such that ∆ ⊆ Σ(R).
We can rephrase our last theorem as follows:
Theorem
Given an ordered stair system (P,≤) in Rd , the simplicial complex
∆(P,≤) has Dushnik-Miller dimension at most d + 1.
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The reciprocal is also true:
Theorem
Given a simplicial complex ∆, there exists an ordered stair system (P,≤)such that ∆(P,≤) = ∆ i� there exists a (d + 1)-representation R such
that ∆ = Σ(R).
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Can we realize any simplicial complex ∆ of Dushnik-Miller dimension at
most d as a contact complex of stairs?
dimDM ≤ 3
?
Yes! The idea is to use the previous theorem and to cut the unnecessary
parts of the stairs.
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Can we realize any simplicial complex ∆ of Dushnik-Miller dimension at
most d as a contact complex of stairs?
dimDM ≤ 3
?
Yes! The idea is to use the previous theorem and to cut the unnecessary
parts of the stairs.
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Consider a simplicial complex ∆ which is included in a Σ(R) where R is a
(d + 1)-representation.
R =≤1 b e d c a
≤2 e c d a b
≤3 a b c d e
Embed the elements in Rd as follows: for any element consider the point
(v1, . . . , vd ) where vi is the position of v in the order ≤i of R .
b e d c a
e
c
d
a
b b
a
d
c
e
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Consider a simplicial complex ∆ which is included in a Σ(R) where R is a
(d + 1)-representation.
R =≤1 b e d c a
≤2 e c d a b
≤3 a b c d e
Embed the elements in Rd as follows: for any element consider the point
(v1, . . . , vd ) where vi is the position of v in the order ≤i of R .
b e d c a
e
c
d
a
b b
a
d
c
e
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Consider a simplicial complex ∆ which is included in a Σ(R) where R is a
(d + 1)-representation.
R =≤1 b e d c a
≤2 e c d a b
≤3 a b c d e
Use order ≤d+1 in order to de�ne the ordered stairs:
b e d c a
e
c
d
a
b b
a
d
c
e
According to the previous theorem, ∆(P,≤d+1) = Σ(R).22 / 29
For example ∆ = (be), (bd), (ed), (edc) and (ac).
b
a
d
c
e
De�ne now the stairs. For any element v , for any face F containing v , add
the corner (c1, . . . , cd ) where ci = maxx∈F xi .
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For example ∆ = (be), (bd), (ed), (edc) and (ac).
b
a
d
c
e
De�ne now the stairs. For any element v , for any face F containing v , add
the corner (c1, . . . , cd ) where ci = maxx∈F xi .
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For example ∆ = (be), (bd), (ed), (edc) and (ac).
b
a
d
c
e
De�ne now the stairs. For any element v , for any face F containing v , add
the corner (c1, . . . , cd ) where ci = maxx∈F xi .
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For example ∆ = (be), (bd), (ed), (edc) and (ac).
b
a
d
c
e
Check that ∆ is the associated contact complex.
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Theorem
Any DM dimension at most d + 1 simplicial complex is the contact
complex of stairs in Rd .
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Open questions:
In not general position?
A reciprocal to the previous Theorem?
Is it possible to guarantee that the stairs have only one �bend�?
b
a
d
c
e
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Thank you for your attention!
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