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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