The Dushnik-Miller dimension of a poset $\le$ is the minimal number $d$ of
linear extensions $\le_1, \ldots , \le_d$ of $\le$ such that $\le$ is the
intersection of $\le_1, \ldots , \le_d$. Supremum sections are simplicial
complexes introduced by Scarf and are linked to the Dushnik-Miller as follows:
the inclusion poset of a simplicial complex is of Dushnik-Miller dimension at
most $d$ if and only if it is included in a supremum section coming from a
representation of dimension $d$. Collapsibility is a topoligical property of
simplicial complexes which has been introduced by Whitehead and which resembles
to shellability. While Ossona de Mendez proved in that a particular type of
supremum sections are shellable, we show in this article that supremum sections
are in general collapsible thanks to the discrete Morse theory developped by
Forman

Bauer, Balthazar

Isenmann, Lucas

English

arXiv

International audienceThe Dushnik-Miller dimension of a poset $\le$ is the minimal number $d$ of linear extensions $\le_1, \ldots , \le_d$ of $\le$ such that $\le$ is the intersection of $\le_1, \ldots , \le_d$. Supremum sections are simplicial complexes introduced by Scarf and are linked to the Dushnik-Miller as follows: the inclusion poset of a simplicial complex is of Dushnik-Miller dimension at most $d$ if and only if it is included in a supremum section coming from a representation of dimension $d$. Collapsibility is a topoligical property of simplicial complexes which has been introduced by Whitehead and which resembles to shellability. While Ossona de Mendez proved in that a particular type of supremum sections are shellable, we show in this article that supremum sections are in general collapsible thanks to the discrete Morse theory developped by Forman

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HAL Id: hal-01867246https://hal.archives-ouvertes.fr/hal-01867246Submitted on 6 Jun 2019HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.Discrete Morse theory for the collapsibility of supremumsectionsBalthazar Bauer, Lucas IsenmannTo cite this version:Balthazar Bauer, Lucas Isenmann. Discrete Morse theory for the collapsibility of supremum sections.ICGT: International Colloquium on Graph Theory and combinatorics, Jul 2018, Lyon, France. ￿hal-01867246￿Discrete Morse theory for the collapsibility of supremum sectionsBalthazar Bauer — INRIA, DIENS, PSL research, CNRS, Paris, FranceLucas Isenmann — LIRMM, Université de Montpellier, CNRS, Montpellier, FranceAbstractThe Dushnik-Miller dimension of a poset ≤ is the minimal number d of linear extensions≤1, . . . ,≤d of ≤ such that ≤ is the intersection of ≤1, . . . ,≤d. Supremum sections are simplicialcomplexes introduced by Scarf [13] and are linked to the Dushnik-Miller as follows: the inclusionposet of a simplicial complex is of Dushnik-Miller dimension at most d if and only if it isincluded in a supremum section coming from a representation of dimension d. Collapsibilityis a topological property of simplicial complexes which has been introduced by Whitehead [18]and which resembles shellability. While Ossona de Mendez [12] proved that a particular type ofsupremum sections are shellable, we show in this article that supremum sections are in generalcollapsible thanks to the discrete Morse theory developped by Forman [8].1 IntroductionThe order dimension (also known as the Dushnik-Miller dimension) of a poset≤ has been introducedby Dushnik and Miller [4]. It is defined as the minimum number d of linear extensions ≤1, . . . ,≤d of≤ such that ≤ is the intersection of these extensions i.e. ∀x, y ∈ V, x ≤ y ⇐⇒ (∀i ∈ [[1, d]], x ≤i y).See [16] for a comprehensive study of this topic. This notion is important because, for example, ofa theorem of Schnyder [14] which states that a graph is planar if and only if the Dushnik-Millerdimension of the inclusion poset of the associated simplicial complex is at most 3. Representationswere introduced by Scarf [13]. A d-representation on a set V is a set of d linear orders on V .Given R a representation on a set V , we can define a simplicial complex Σ(R) associated to thisrepresentation that we call its supremum section. Scarf proved that every supremum section of arepresentation satisfying some additional properties, so called “standard”, is the inclusion poset ofa d-polytope with one face removed. Ossona de Mendez [12] proved that every abstract simplicialcomplex of Dushnik-Miller dimension at most d is contained in a complex which is shellable andhas a straight line embedding in Rd−1. Supremum sections also appeared in commutative algebra:Bayer et al. [2] studied monomial ideals which are linked to supremum sections by what they callScarf complexes. They are used by Felsner et al. [7] in order to study orthogonal surfaces. They alsoappear in the study of Gonçalves et al. [9] of a variant of Delaunay graphs and in the study of emptyrectangles graphs by Felsner [6]. Furthermore, they also appear in spanning-tree-decompositionsand in the box representations problem as shown by Evans et al. [5].The goal of our article is to generalize the result of Ossona de Mendez about the shellabilityof standard supremum sections to every supremum sections. As there exists supremum sectionswhich are not shellable, for instance the simplicial complex characterized by its facets {a, b, c} and{c, d, e}, we will replace shellability by collapsibility which is a similar notion. A collapse is atopological operation on simplicial complexes, and more generally on CW-complexes, introducedby Whitehead [18] in order to define a simple homotopy equivalence which is a refinement of thehomotopy equivalence. A complex is said to be collapsible if it collapses to a point. See [10]for a comprehensive study of this topic. The discrete Morse theory introduced by Forman [8] isbased on this notion and has numerous applications in applied mathematics and computer science.Homotopy equivalence is a topological notion of topological spaces introduced to classify topologicalarXiv:1803.09577v2  [cs.DM]  4 May 2018spaces. Roughly speaking, two spaces are said to be homotopy equivalent if there exists a continuousdeformation from one to the other. A topological space is said to be contractible if it is homotopyequivalent to a point. Collapsible spaces form an important subclass of contractible spaces. Whilecontractibility is algorithmically undecidable by a result of Novikov [17], the subclass of collapsiblespaces is algorithmically recognizable. More precisely Tancer [15] showed that it is NP-complete todecide whether a simplicial complex is collapsible. Furthermore, every 1-dimensional contractiblecomplex is collapsible but the house with two rooms [1] and the dunce hat [19] show that there arecomplexes which are contractible but not collapsible. Finally, the conjecture of Zeeman [19], whichimplies the Poincarré conjecture, states that for every finite contractible 2-dimensional CW-complexK, the space K × [0, 1] is collapsible.2 NotationsIn the following, V is a finite set. An (abstract) simplicial complex ∆ is a subset of P(V ) closed byinclusion (i.e. ∀X ∈ ∆,∀Y ⊆ X,Y ∈ ∆). We call faces the elements of ∆ and facets the maximalfaces of ∆ according to the inclusion order.Definition 1 (Ossona de Mendez [12]). Given a linear order ≤ on a set V , an element x ∈ V ,and a set F ⊆ V , we say that x dominates F in ≤, and we denote it F ≤ x, if f ≤ x for everyf ∈ F . A d-representation R on a set V is a set of d linear orders ≤1, . . . ,≤d on V . Given ad-representation R, an element x ∈ V , and a set F ⊆ V , we say that x dominates F in R if xdominates F in some order ≤i∈ R. We define Σ(R) as the set of subsets F of V such that everyv ∈ V dominates F in R. The set Σ(R) is called the supremum section of R.It is easy to show that if R is a d-representation on a set V , then Σ(R) is a simplicial complex.An example is the following 3-representation on {a, b, c, d, e}: a <1 b <1 e <1 d <1 c, c <2 b <2a <2 d <2 e, and e <3 d <3 c <3 b <3 a. The corresponding complex Σ(R), depicted on the left ofFigure 1, is characterized by its facets {a, b}, {b, c, d}, and {b, d, e}. For example {a, b, c} is not inΣ(R) as b does not dominate {a, b, c} in any order.Definition 2. Let ∆ be a simplicial complex. We say that a face F of ∆ is a free face of ∆ if itis non-empty, non-maximal and contained in only one facet of ∆.Let ∆ and Γ be two simplicial complexes. We say that ∆ collapses to Γ if there exists ksimplicial complexes ∆1, . . . ,∆k and a free face Fi of ∆i for every i ∈ [[1, k − 1]] such that ∆1 = ∆, ∆i+1 = ∆i \ {F ∈ ∆i : Fi ⊆ F} for every i ∈ [[1, k− 1]] and ∆k = Γ. We say that ∆ is collapsibleif it collapses to a point.The Hasse diagram of a poset is the transitive reduction of the digraph of the poset. Let R bea representation on a set V , we denote H(R) the Hasse diagram of the inclusion poset of Σ(R).Definition 3. Let (≤, V ) be a poset and let M be a matching of the Hasse diagram of ≤. For an arca of the Hasse diagram of ≤, we denote d(a) and u(a) the elements of V such that a = (u(a), d(a))and d(a) < u(a). A matching M of the Hasse diagram of ≤ is said to be acyclic if, when reversingthe orientation of the arcs of M , the Hasse diagram remains acyclic.It is known that if ≤ is the poset of inclusion of a simplicial complex and M is a matching of theHasse diagram of ≤ then M is acyclic if and only if there is no sequence of arcs m1, . . . ,mn of Msuch that (u(mi+1), d(mi)) is in the Hasse diagram for all i ∈ [[1, n− 1]] as well as (u(m1), d(mn)).Theorem 4 (Chari [3]). Let ∆ be a simplicial complex. If the Hasse diagram of the inclusion posetof ∆ admits a complete ( i.e. perfect) acyclic matching, then ∆ is collapsible.3 Our contributionTheorem 5. Let R be a representation on a set V . Then Σ(R) is collapsible.Because of Theorem 4, it is enough to show that if R is a representation on a set V , then H(R)admits a complete acyclic matching.3.1 ProofsThe proof relies on an induction on the dimension of the representation R. Let R be a d-representation on a set V . We denote R′ = (≤1, . . . ,≤d−1) the (d−1)-representation on V obtainedfrom R by deleting the order ≤d.Lemma 6. The simplicial complex Σ(R′) is a subcomplex of Σ(R).Proof. Let F be a face of Σ(R′). As every element x of V dominates F in at least one of the orders≤1, . . . ,≤d−1, the element x also dominates F in at least one of the orders ≤1, . . . ,≤d. We concludethat F ∈ Σ(R).See Figure 1 to see what Σ(R′) is for the example representation of Definition 1.a becda bca bcd eab bc dc bd be debcd bde∅Figure 1: R denotes the representation from the example. From left to right: Σ(R) where a greysection corresponds to a face with 3 elements, Σ(R′) where R′ is the representation obtained from Rby deleting the order ≤3, the Hasse diagram of Σ(R) where the crossed-out faces are the faces fromΣ(R) \Σ(R′) and where the fat edges correspond to a complete acyclic matching of Σ(R) \Σ(R′).Lemma 7. We define the function ψ byψ : Σ(R) \ Σ(R′) → VF 7→ min<d {x ∈ V : x <i (max≤i F ) ∀i ∈ [[1, d− 1]]}Then the function ψ is well-defined.Proof. Let F be a face of Σ(R)\Σ(R′). We denote fi = max≤i F for every i ∈ [[1, d]]. As F 6∈ Σ(R′),there exists an element x ∈ V such that ∀i ∈ [[1, d − 1]], x <i fi. So the minimum is taken in anon-empty set.We define the sets A = {F ∈ Σ(R)\Σ(R′) : ψ(F ) 6∈ F} and B = {F ∈ Σ(R)\Σ(R′) : ψ(F ) ∈ F}.The goal is to find a complete acyclic matching between A and B.Lemma 8. For every F ∈ A, we have F ∪ {ψ(F )} ∈ B, ψ(F ∪ {ψ(F )}) = ψ(F ) and max≤d F <ψ(F ).For every F ∈ B, we have F \ {ψ(F )} ∈ A and ψ(F \ {ψ(F )}) = ψ(F ).Proof. Let F be in A, we denote F ′ = F⋃{ψ(F )}. For every i ∈ [[1, d]], we denote fi (resp.f ′i) the maximum of F (resp. F′) in the order ≤i. By definition of ψ, ψ(F ) <i fi for everyi ∈ [[1, d − 1]]. Furthermore, fd <d ψ(F ), otherwise ψ(F ) would not dominate F . Thus, fi = f ′ifor every i ∈ [[1, d − 1]] and fd <d f ′d = ψ(F ). Suppose that F ′ 6∈ Σ(R). Then there would exista such that a does not dominate F ′ in any order. Thus a <i fi(= f′i) for every i ∈ [[1, d − 1]] anda <d ψ(F ) which contradicts the minimality of ψ(F ). We deduce that F′ ∈ Σ(R).As ψ(F ) <i f′i for every i ∈ [[1, d − 1]], ψ(F ) does not dominate F ′ in R′, F ′ 6∈ Σ(R′) andψ(F ′) ≤d ψ(F ). If ψ(F ′) <d ψ(F ) then we would have ψ(F ′) <i f ′i for every i ∈ [[1, d]] asf ′d = ψ(F ). We deduce that ψ(F′) = ψ(F ) = f ′d ∈ F ′. Finally, we conclude that F ∪ {ψ(F )} ∈ B.The second property can be proved in the same manner.Lemma 9. The Hasse diagram of the inclusion poset of Σ(R) \ Σ(R′) admits a complete acyclicmatching.See Figure 1 to see an example of a complete acyclic matching.Proof. We define the function ϕ : A → B defined by ϕ(F ) = F ∪ {ψ(F )} for every F ∈ A. Letus show that ϕ is a bijection. To do so, we define the function η : B → A by η(F ) = F \ {ψ(F )}where F ∈ B. Lemma 8 implies that η is well defined, that η ◦ϕ = idA, and that ϕ ◦ η = idB. Thusϕ is a bijection and ϕ defines a complete matching M = {(F,ϕ(F )) : F ∈ A} between A and B.Suppose that M is not acyclic: there exists a sequence m1, . . . ,mn of arcs of M where mi =(Fi ∪ {ψ(Fi)}, Fi) for a Fi ∈ A for every i ∈ [[1, n]] such that (Fi+1 ∪ {ψ(Fi+1)}), Fi) is in theHasse diagram for every i ∈ [[1, n − 1]] as well as (F1 ∪ {ψ(F1)}, Fn). As for every i ∈ [[1, n − 1]],Fi ⊆ Fi+1 ∪ ψ(Fi+1) and |Fi| + 1 = |Fi+1 ∪ ψ(Fi+1)|, we deduce that ψ(Fi+1) ∈ Fi. Thereforeψ(Fi+1) <d ψ(Fi) for every i ∈ [[1, n − 1]] and thus ψ(Fn) <d ψ(F1). As (F1 ∪ {ψ(F1)}, Fn) is inthe Hasse diagram, we show in the same way that ψ(F1) <d ψ(Fn) which contradicts the fact thatψ(Fn) <d ψ(F1). We conclude that M is a perfect acyclic matching of Σ(R) \ Σ(R′).We can now prove Theorem 5.Proof. We prove the result by induction on the number of orders. LetR = (≤1) be a 1-representationon V . We denote m1 the minimum on V in ≤1. Let F be a face of Σ(R) which contains an elementx different from m1. Then m1 does not dominate F in R as m1 <1 x. The set {m1} is a face ofΣ(R) as every element of V dominates {m1} in the order ≤1. Thus Σ(R) = {∅, {m1}} and H(R)admits a complete acyclic matching ({m1}, ∅). The base case is therefore true.Let d ≥ 2, we now suppose that the result is true for any (d− 1)-representation on V . Let R =(≤1, . . . ,≤d) be a d-representation on V . We denote R′ = (≤1, . . . ,≤d−1) the (d−1)-representationon V obtained from R by deleting the order ≤d. We define K as Σ(R)\Σ(R′). Because of Lemma 9,the Hasse diagram of the inclusion poset of K admits a complete acyclic matching M1.By induction hypothesis, H(R′) admits a complete acyclic matching M2. Thus M1 ∪M2 is acomplete matching of H(R). Furthermore, H(R) is the union of H(R′) and the Hasse diagram ofthe inclusion poset of K with some arcs between K and Σ(R′). If an arc between K and Σ(R′)is oriented from Σ(R′) to K, then there would be a face F of Σ(R′) that would contain a face Gof K. As Σ(R′) is closed by inclusion, then G is also in Σ(R′) which contradicts the definition ofK. Therefore the arcs between K and Σ(R′) are oriented from K to Σ(R′) and we deduce thatM1 ∪M2 is a perfect acyclic matching of H(R). We conclude by induction.References[1] R.H. Bing, Some aspects of the topology of 3-manifolds related to the Poincaré conjecture, Lectures onModern Mathematics , II , Wiley, pp. 93–128, 1964.[2] D. Bayer, I. Peeva, B. Sturmfels, Monomial resolutions, Mathematical Research Letters 5, 31 – 46, 1998.[3] M. K. Chari, On discrete Morse functions and combinatorial decompositions. Discrete Mathematics,217(1-3), 101-113, 2000.[4] B. Dushnik and E.W. Miller, Partially ordered sets, American Journal of Mathematics, 63(3): 600-610,1941.[5] W. Evans, S. Felsner, G. Kobourov, and T. 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Scarf, The Computation of Economic Equilibria, vol. 24 of Cowles Foundation Monograph, YaleUniversity Press, 1973.[14] W. Schnyder, Planar graphs and poset dimension, Order 5(4): 323-343, 1989.[15] M. Tancer, Recognition of collapsible complexes is NP-complete, Discrete and Computational Geometry,55(1), 21-38, 2016.[16] W. T. Trotter. Combinatorics and partially ordered sets: Dimension theory. Johns Hopkins Series inthe Mathematical Sciences, The Johns Hopkins University Press, 1992.[17] I. A. Volodin, V. E. Kuznetsov and A. T. Fomenko. The problem of discriminating algorithmically thestandard three-dimensional sphere. Russian Mathematical Surveys, 29(5), 71-172, 1974.[18] J. H. C. Whitehead, Simplicial Spaces, Nuclei and m-groups. Proceedings of the London mathematicalsociety, 2(1), 243-327, 1939.[19] E. C. Zeeman, On the dunce hat. Topology, vol. 2, no 4, p. 341-358, 1963.

Discrete Morse theory for the collapsibility of supremum sections

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