Foldings in graphs and relations with simplicial complexes and posets
aa r X i v : . [ m a t h . C O ] O c t Foldings in graphs and relations with simplicial complexes and posets
Etienne Fieux , Jacqueline Lacaze Universit´e Paul Sabatier, Toulouse, France
Abstract
We study dismantlability in graphs. In order to compare this notion to similar operations in posets(partially ordered sets) or in simplicial complexes, we prove that a graph G dismants on a subgraph H ifand only if H is a strong deformation retract of G . Then, by looking at a triangle relating graphs, posetsand simplicial complexes, we get a precise correspondence of the various notions of dismantlability ineach framework. As an application, we study the link between the graph of morphisms from a graph G to a graph H and the polyhedral complex Hom( G, H ); this gives a more precise statement about wellknown results concerning the polyhedral complex Hom(
G, H ) and its relation with foldings in G or H . Keywords : dismantlability; foldings; Hom complex; posets; simplicial complexes; strong deformationretract
A vertex g of a graph G is said dismantlable if there is another vertex a in G such that N G ( x ) ⊂ N G ( a ) where N G ( x ) := { y ∈ V ( G ) , y ∼ x } is the open neighborhood of x . This will be denoted x ⊢ d a and we will also saythat a dominates x . The passage from G to G − x by deleting a dismantlable vertex x is called a folding anddenoted G ց d G − x ; the resultant graph G − x is called a fold of G . A succession of foldings will be calleda dismantling . If there is a dismantling from a graph G to a subgraph H , we say that G is dismantlable on and write G ց d H ; this means that there is a dismantling sequence x , . . . , x k from G to H , i.e. V ( G ) = V ( H ) ∪ { x , . . . , x k } with x i dismantlable in the subgraph induced by V ( H ) ∪ { x i , x i +1 , . . . , . . . , x k } for i = 1 , , . . . , k ; this will be also denoted H d ր G . A reflexive graph G is said dismantlable if it is dismantlableon a looped vertex. Following [HN04], a graph whose every vertex is non dismantlable is called stiff .It seems that the the first papers which focused on vertices whose open neighborhood is included in theopen neighborhood of another vertex are [Qui83] and [NW83]where it was proved independently that areflexive graph is cop win if, and only if, it is dismantlable. The reflexive bridged and connected graphs (agraph is bridged if it contains no isometric cycles of length greater than three; in particular, the chordalgraphs are bridged) are examples of dismantlable graphs ([AF88]). In this paper, the objective is to givea precise description of the relation between dismantlability in graphs and similar operations in partiallyordered sets (posets) or in simplicial complexes. In section 2, we give a characterization of foldings anddismantlings by the way of morphisms and homotopies. The key result (Proposition 2.2) is that a graph G dismants on a subgraph H if, and only if, H is a strong deformation retract of G . As a useful corollary,we get that if G ′ and G ′′ are two subgraphs of a graphs G such that G ′′ is a subgraph of G ′ , G ց d G ′ and G ց d G ′′ then we can conclude that G ′ ց d G ′′ (Corollary 2.1).In the framework of posets, there is also a very well known notion of dismantlability (most frequentlynamed irreducibility ; see Section 3 for a brief discussion). From the seminal paper [Sto66], we know thatthe dismantlings in posets allow to describe the homotopy type of a poset (its real homotopy type, i.e.the homotopy type of the poset considered as a topological space and not the homotopy type of its ordercomplex). Dismantlability in posets has been studied in various articles, in particular in relation with thefixed point property ([BB79],[Riv76],[Sch03],[Wal84]). It is known ([BCF94],[Gin94]) that the dismantlabilityof a poset P is equivalent to the dismantlability of its comparibility graph (which will be called Comp ( P )).In [Gin94], it was also proved that the dismantlability of a graph G is equivalent to the dismantlability of the poset of complete subgraphs of G (which will be called C ( G )). In section 3, we will give a generalization of Institut de Math´ematiques de Toulouse, Universit´e Paul Sabatier, 118 Route de Narbonne, 31062 Toulouse Cedex 09,France ; fi[email protected] Institut de Recherche en Informatique de Toulouse, Universit´e Paul Sabatier, 118 Route de Narbonne, 31062 ToulouseCedex 09, France ; [email protected] It is important to note that several papers (as [AF88],[BFJ08],[Gin94],[LPVF08],[Qui83]) take another definition of dis-mantlability : a vertex x is dismantlable if there is another vertex a such that N G [ x ] ⊂ N G [ a ] where N G [ x ] := N G ( x ) ∪ { x } isthe closed neighborhood of x . Of course, the two definitions are the same when the graphs are reflexive. K of finite simplicial complexes. We show in section 4 that it hh corresponds ii to thedismantlability in graphs under natural functors relating G and K .So, this gives a hh good ii behaviour of a triangle ( G ◦ , P , K ) in relation to the various notions ofdismantlability in G ◦ , P or K and, consequently, with the equivalences classes (named homotopy classes )defined by the operation of dismantlability (section 5). A motivation for this question is given by thepolyhedral complex Hom ( G, H ) associated to two graphs G and H . This construction is due to Lovasz afterits pioneering work ([Lov78]) where he solved the Kneser conjecture by using the simplicial complex N ( G ),the neighborhood complex of G . Since the article [BK06] (where the authors proved in particular that Hom ( K , G ) and N ( G ) have the same homotopy type), the Hom complex has became an important tool fordetermining lower bounds to the chromatic number of certain graphs (see [Koz08] for a complete expositionand more references). For obtaining topological information about the polyhedral complex
Hom ( G, H )(which is not in general a simplicial complex), it is usual to look at its face poset F P ( Hom ( G, H )) or at theorder complex of its face poset, i.e. its barycentric subdivision Bd ( Hom ( G, H )) = ∆ P ( F P ( Hom ( G, H )))(which is a simplicial complex). On the other hand, the set of morphisms from G to H is the vertex set of agraph (called hom G ( G, H )) and is also the vertex set of
Hom ( G, H ); we will study the relation between thegraph hom G ( G, H ) and the polyhedral complex
Hom ( G, H ) by using the triangle ( G ◦ , P , K ) and regardingthem in P (Proposition 6.1). In particular, this gives another proof of a result describing the dismantlingson Hom ( G, H ) induced by foldings on G or H . However, this result which is usually formulated in terms ofsimplicial complexes is formulated here in terms of graphs. Notations
In this paper, the graphs will be finite, undirected and without parallel edges. The vertex setof a graph G is denoted V ( G ). The set of these graphs will be denoted G and eventually considered as acategory where a morphism f : G → G ′ from a graph G to a graph G ′ is an application from V ( G ) to V ( G ′ )which preserves adjacency ( x ∼ y = ⇒ f ( x ) ∼ f ( y )); G ◦ will denote the subcategory obtained by retrictingto reflexive graphs (i.e., graphs G such that x ∼ x for all x in V ( G )).Let G ∈ G . If X is a subset of V ( G ), the notation G − X will indicate the subgraph of G induced bythe set of vertices V ( G ) \ X . In particular, if x ∈ V ( G ), G − x will be an abbreviated form of G − { x } and i x : G − x → G will denote the inclusion morphism. If x ⊢ d a , the folding G → G − x which sends x to a (andis the identity on G − x ) will be denoted r x,a .The notation G d ր G + y means that we have a added a vertex y to G in such a way that y is dismantlablein the new graph. In this section, we characterize foldings and dismantlings in terms of morphisms. Let
G, G ′ ∈ G . The set ofmorphisms from G to G ′ is the vertex set of a graph, denoted hom G ( G, G ′ ), where f ∼ f ′ in hom G ( G, G ′ )if and only if x ∼ y in G implies f ( x ) ∼ f ′ ( y ) in G ′ ([HHMNL88],[BCF94]); this graph is reflexive because f ∼ f means precisely that f is a morphism of graph. By an abuse of notation, hh f ∈ hom G ( G, G ′ ) ii willmean that f is a morphism from G to G ′ (in place of f ∈ V ( hom G ( G, G ′ ))). Remark 2.1
Let
G, G ′ , G ′′ ∈ G , f, f ′ ∈ hom G ( G, G ′ ) and h, h ′ ∈ hom G ( G ′ , G ′′ ) . If f ∼ f ′ and h ∼ h ′ , then h ◦ f ∼ h ′ ◦ f ′ because x ∼ y in G implies f ( x ) ∼ f ′ ( y ) in G ′ (by f ∼ f ′ ) which implies h ◦ f ( x ) ∼ h ′ ◦ f ′ ( y ) in G ′ (by h ∼ h ′ ). An important class of morphisms is given by retractions. A retraction of a graph G to a subgraph H of G isa morphism r : G → H such that r ( x ) = x for all x in V ( H ). So, a morphism r : G → G such that r ◦ r = r is a retraction of G to r ( G ). The results of this paragraph are based on the following remarks: Remark 2.2 a. Let f ∈ hom G ( G, G ) . If f ∼ G (where G is the identity morphism on G ), then everyvertex x of G verifies either f ( x ) = x , or x ⊢ d f ( x ) .b. In particular, if f : G → G is a retraction such that f ∼ G , then f ց d f ( G ) . We note that Remark 2.2.a. implies that 1 G is an isolated vertex in hom G ( G, G ) when G is a stiff graph(this is a classical result used in [BCF94], [Doc09]). By definition, a folding is a retraction G → G − x whichsends x to a vertex a which dominates x . However, a general retraction G → G − x is not necessarily afolding (see Figure 1) 2 a • x •• Figure 1: The retraction G → G − x (which sends x to a ) is not a foldingFrom Remark 2.2.a, we get the following characterization of foldings: Lemma 2.1
Let G ∈ G , x ∈ V ( G ) and f : G → G − x a retraction; the following assertions are equivalent:1. f is a folding (i.e., x ⊢ d f ( x ) )2. i x ◦ f ∼ G (where i x is the inclusion G − x ֒ → G ) We conclude also from Remark 2.2.b that foldings on graphs induce dismantlability in graphs of morphisms:
Proposition 2.1
1. If x is dismantlable in G , then hom G ( G, H ) ց d hom G ( G − x, H ) (by identifying hom G ( G − x, H ) with an induced subgraph of hom G ( G, H ) ).2. If x is dismantlable in H , then hom G ( G, H ) ց d hom G ( G, H − x ) (by identifying hom G ( G, H − x ) withan induced subgraph of hom G ( G, H ) ). Proof :
1. Let x dismantlable in G with x ⊢ d a . Then, the map Ψ x,a : hom G ( G − x, H ) → hom G ( G, H ) definedby Ψ x,a ( f ) = f ◦ r x,a is an injective morphism of graphs and we identify hom G ( G − x, H ) with the subgraphΨ x,a ( hom G ( G − x, H )) of hom G ( G, H ). Let us denote Φ x : hom G ( G, H ) → hom G ( G − x, H ) the restrictionmorphism defined by Φ x ( f ) = f ◦ i x ≡ f | G − x . If f ∈ hom G ( G − x, H ), then (Φ x ◦ Ψ x,a )( f ) = ( f ◦ r x,a ) ◦ i x = f ◦ ( r x,a ◦ i x ) = f ; so Φ x ◦ Ψ x,a = 1 hom G ( G − x,H ) and this means that Ψ x,a ◦ Φ x : hom G ( G, H ) → hom G ( G, H )is a retraction to hom G ( G − x, H ) identified with Ψ x,a ( hom G ( G − x, H )). If f ∈ hom G ( G, H ), Ψ x,a ◦ Φ x ( f )takes the same value as f on vertices distinct from x and takes the value f ( a ) on x . Let f, f ′ ∈ hom G ( G, H )with f ∼ f ′ . As i x ◦ r x,a ∼ G (Lemma 2.1), we have f ◦ i x ◦ r x,a ∼ f ′ by Remark 2.1 ; this proves thatΨ x,a ◦ Φ x ∼ hom G ( G,H ) and we conclude hom G ( G, H ) ց d hom G ( G − x, H ) by Remark 2.2.b.2. Similarly, if x is dismantlable in H with x ⊢ d b , we denote Φ x,b : hom G ( G, H ) → hom G ( G, H − x )the morphism of graphs defined by Φ x,b ( f ) = r x,b ◦ f and we identify hom G ( G, H − x ) with the inducedsubgraph of hom G ( G, H ) given by its image under the injection Ψ x : hom G ( G, H − x ) → hom G ( G, H )defined by Ψ x ( f ) = i x ◦ f . Then Φ x,b ◦ Ψ x = 1 hom G ( G,H − x ) and Ψ x ◦ Φ x,b : hom G ( G, H ) → hom G ( G, H ) is aretraction to hom G ( G, H − x ) identified with Ψ x ( hom G ( G, H − x )). If f ∈ hom G ( G, H ), Ψ x ◦ Φ x,b ( f ) takesat a vertex z the same value as f when f ( z ) = x and the value b when f ( z ) = x . It is easy to verify thatΨ x ◦ Φ x,b ∼ hom G ( G,H ) and this proves hom G ( G, H ) ց d hom G ( G, H − x ). (cid:3) Morphisms give rise to a notion of homotopy and it was noticed in [Qui83] that a graph is dismantlable ifand only if the identity morphism is homotopic to a constant morphism. Following [Doc09], for N ∈ N ∗ , I N is the reflexive graph with looped vertices 0 , , , . . . , N and adjacencies 0 ∼ ∼ ∼ ∼ . . . ∼ N − ∼ N . • • • • • • I If f, f ′ ∈ hom G ( G, G ′ ), a homotopy from f to f ′ is a morphism of graphs H : I N → hom G ( G, G ′ ) , i
7→ H i such that H = f and H N = f ′ ; this will be denoted f ≃ f ′ and this means that f and f ′ are in the sameconnected component of hom G ( G, G ′ ). A subgraph G ′ of G is a strong deformation retract if there is ahomotopy H : I N → hom G ( G, G ) such that H = 1 G , H i | G ′ = 1 G ′ for all i ∈ { , , . . . , N } and H N : G → G is actually a retraction to G ′ . The following results will be useful in the sequel:3 emma 2.2 Let G ′′ ⊂ G ′ ⊂ G inclusions of graphs.1. If G ′′ is a strong deformation retract of G ′ and G ′ is a strong deformation retract of G , then G ′′ isa strong deformation retract of G .2. If G ′′ is a strong deformation retract of G and G ′ a retract of G , then G ′′ is a strong deformationretract of G ′ . Proof :
1. Straightforward. 2. Let i G ′ : G ′ ֒ → G the inclusion and r G ′ : G → G ′ a retraction of G to G ′ . If H : I N → hom G ( G, G ) is a homotopy proving that G ′′ is a strong deformation retract of G , then H ′ : I N → hom G ( G ′ , G ′ ) defined by H ′ i = r G ′ ◦H i ◦ i G ′ is a homotopy proving that G ′′ is a strong deformationretract of G ′ . (cid:3) By Lemma 2.1, a fold G − x of G is a strong deformation retract of G ; more generally, dismantlability ischaracterized by strong deformations: Proposition 2.2
Let H be a subgraph of a graph G . Then, G ց d H if, and only if, H is a strong deformationretract of G . Proof :
Let us suppose that G ց d H . This means that one can go from G to H by a composition offoldings ; each fold being a strong deformation retract, H is a strong deformation retract of G by Lemma2.2 a. If we suppose now that H is a strong deformation retract of G , we can use an argument similar tothat used in the proof of Th´eor`eme 4.4 in [BCF94]. Let H : I N → hom G ( G, G ) be a homotopy provingthat H is a strong deformation retract of G . If H = G , H N = 1 G , and we can suppose H = 1 G . If a ∈ G is such that H ( a ) = a , then a / ∈ H , a ⊢ d H ( a ) in G and G ց d G − a . For i ∈ { , , . . . , N } , define H ′ i : G − a → G − a by H ′ i ( x ) = ( r a, H ( a ) ◦ H i )( x ) for x ∈ G − a . Clearly, these morphisms define a homotopy H ′ : I N → hom G ( G − a, G − a ), H ′ ∼ G − a and H ′ N is a retraction from G − a to H . Thus, H is a strongdeformation retract of G − a . If G − a = H , we can iterate, and so G ց d H . (cid:3) Corollary 2.1
Let G ′′ ⊂ G ′ ⊂ G inclusions of graphs such that G ց d G ′ and G ց d G ′′ . Then G ′ ց d G ′′ . Proof :
Straightforward from Lemma 2.2 b. and Proposition 2.2. (cid:3)
Let us recall that two graphs G and H are homotopically equivalent if there is f ∈ hom G ( G, H ) and g ∈ hom G ( H, G ) such that g ◦ f ≃ G and f ◦ g ≃ H . In particular, if H is a strong deformation retract of G , H and G are homotopically equivalent. We mention the following well known result (two quite differentproofs are given in [BCF94] and [HN04]): Proposition 2.3
Let G ∈ G and H, H ′ two stiff subgraphs such that G ց d H and G ց d H ′ . Then H isisomorphic to H ′ . Proof :
By Proposition 2.2, H and H ′ are strong deformation retracts of G . So, H and H ′ are homotopicallyequivalent. Let f ∈ hom G ( H, H ′ ) and g ∈ hom G ( H ′ , H ) such that g ◦ f ≃ H and f ◦ g ≃ H ′ . As the graphs H and H ′ are stiff, the connected components of 1 G and 1 H are reduced, respectively, to { G } and { H } .So, we conclude that g ◦ f = 1 H and f ◦ g = 1 H ′ and that H and H ′ are isomorphic. (cid:3) ( G , P ) Let P the category of finite posets. If P, Q ∈ P , a morphism of posets f : P → Q is a map from P to Q which preserves the order (i.e. x ≤ y in P implies f ( x ) ≤ f ( y ) in Q ). An element p of a poset P willbe called dismantlable if either P >p := { y ∈ P, y > p } has a least element or P
p , p will be said dominated by a . Thedeletion of the dismantlable element x , will be denoted P ց d P \ { x } and P ց d Q means that one can gofrom the poset P to a subposet Q by successive deletions of dismantlable elements. Proposition 3.1
Let f : P → P a morphism of posets map such that either f ≤ P or f ≥ P . Then P ց d Fix ( f ) where Fix ( f ) := { p ∈ P, f ( p ) = p } . roof : We suppose that Fix ( f ) = P (i.e., f = 1 P ) and we consider the case f ≤ P . Let x minimal in P \ Fix( f ). Let y < x (for example, y = f ( x )). Then y = f ( y ) (by minimality of x in P \ Fix( f )) and y ≤ f ( x )(because y < x ⇒ f ( y ) ≤ f ( x )). Thus, f ( x ) is the greatest element of P This proof is essentially the proof given by Kozlov in the particular case f = f ([Koz06,Theorem 2.1] or [Koz08, Theorem 13.12], where the conclusion is given in terms of simplicial complexes). Comp : P → G ◦ Let P ∈ P . The comparability graph of P , denoted Comp ( P ), is the graph whose vertex set is P withadjacencies x ∼ y if and only if x and y are comparable in P . In particular, for every poset P , Comp ( P ) isa reflexive graph. We will say that a graph G is a cone with apex a if y ∼ a for all y ∈ V ( G ) (this definitionimplies that the apex is a looped vertex). The following facts are easy : • If x is a looped vertex of a graph G , then x is dismantlable if, and only if, N G ( x ) − x is a cone. • Comp ( P ) − x = Comp ( P \ { x } ). • N Comp ( P ) ( x ) − x = Comp ( P >x ∪ P Let P, Q ∈ P . If P ց d Q , then Comp ( P ) ց d Comp ( Q ) . Proof : Clearly, if x is dominated by an element a in P , then x is dominated by the vertex a in Comp ( P ).Consequently, P ց d P \ { x } = ⇒ Comp ( P ) ց d Comp ( P ) − x = Comp ( P \ { x } ) and the proposition followsby iteration. (cid:3) Reciprocally, Comp ( P ) ց d Comp ( Q ) doesn’t imply in general P ց d Q because a dismantlable vertex in Comp ( P ) is not necessarily a dismantlable element in P (see, for example, the poset P = { a, b, c, d } with d < b, c < a given in Figure 3). • b • c • a • dP • b • c • a • dComp ( P )Figure 3: a and d are non dismantlable in P and dismantlable in Comp ( P )A poset will be called a double cone with apex a if it admits an element a comparable with all elementsof the poset. It is clear that P is a double cone if, and only if, Comp ( P ) is a cone and this motivates thefollowing definition. D´efinition 3.1 An element p of a poset P is said weakly dominated by a if P >p ∪ P
5n other terms, p is weakly dominated by a if p and a are comparable and if every element comparablewith p is also comparable with a . Of course, if p is dominated by a , then p is weakly dominated by a butthe reverse implication is false in general (in the poset P given in Figure 3, d is weakly dominated by a butis not dominated by a ). Let p an element of a poset P ; the following assertions are equivalents :1) p is dominated by a in Comp ( P )2) N Comp ( P ) ( p ) − p is a cone with apex a Comp ( P >p ∪ P p ∪ P
Let P, Q ∈ P . Then, P ց wd Q ⇐⇒ Comp ( P ) ց d Comp ( Q ) C : G → P Let G ∈ G . We recall that a complete subgraph H of G is an induced subraph of G such that x ∼ y for anydistinct vertices x and y of H ; a complete subgraph of G will be identified with its set of vertices. The posetof complete subgraphs of G , denoted C ( G ), is the poset given by the set of non empty complete subgraphsof G with the inclusion as order relation. Theorem 3.2 Let G ∈ G and H a subgraph of G such that G ց d H . If all vertices in V ( G ) \ V ( H ) arelooped, then C ( G ) ց d C ( H ) .In particular, if G ∈ G ◦ and H a subgraph of G , then G ց d H = ⇒ C ( G ) ց d C ( H ) . Proof : Let x ∈ V ( G ) \ V ( H ) be a dismantlable and looped vertex with a which dominates x . We define f : C ( G ) → C ( G ) by f ( c ) = c ∪ { a } if x ∈ c and f ( c ) = c if x c ; note that f is well defined because x is looped. Then f ≥ C ( G ) and, by Proposition 3.1, C ( G ) ց d Im( f ). Now, let f : Im( f ) → Im( f )defined by f ( c ) = c \ { x } if x ∈ c and f ( c ) = c if x c . Then f ≤ Im( f ) and, by Proposition 3.1,Im( f ) ց d Im( f ) = C ( G − x ). So, C ( G ) ց d C ( G − x ) and the proposition follows by iterating the process. (cid:3) Now, before studying the reciprocal of Theorem 3.2, we recall that an element p of a poset P is an atomif P
For all P ∈ P , RU B ( P ) ց d m ( P ) . Proof : As m ( P ) is the subgraph of RU B ( P ) induced by the set of atoms A ( P ), it suffices to prove thatevery vertex in V ( RU B ( P )) \ V ( m ( P )) (i.e., every element of P which is not an atom) is dominated by avertex of m ( P ). Let q ∈ V ( RU B ( P )) \ V ( m ( P )) = P \ A ( P ). It is immediate that q ⊢ d x for every vertex x ∈ V ( m ( P )) = A ( P ) such that x < q (because z ∼ q ⇐⇒ P ≥ z,q = ∅ = ⇒ P ≥ z,x = ∅ = ⇒ z ∼ x ). (cid:3) Proposition 3.4 Let P in P and x dismantlable in P . Then, RU B ( P ) ց d RU B ( P \ { x } ) .As a consequence, P ց d Q = ⇒ RU B ( P ) ց d RU B ( Q ) . Proof : Let us suppose that x is dominated by a in P . First, we verify that x is dominated by a in RU B ( P ). So, let y ∈ P such that y ∼ x in RU B ( P ). If y = x , then x ∼ a in RU B ( P ) because P ≥ y,a = P ≥ x,a = ∅ . If y = x and z ∈ P ≥ y,x , then z ∈ P ≥ y,a (because z ≥ x and x is dominated by a ); so, y ∼ a and x is also dominated by a in RU B ( P ). Hence we have RU B ( P ) ց d RU B ( P ) − x . Now,we compare the graphs RU B ( P ) − x and RU B ( P \ { x } ). They have the same vertex sets and clearly RU B ( P \ { x } ) is a subgraph of RU B ( P ) − x (if P ≥ y,z = ∅ in P \ { x } , we have also P ≥ y,z = ∅ in P ).6ow, let us suppose that y ∼ z in RU B ( P ) − x ; this means that P ≥ y,z = ∅ . If x is in P ≥ y,z , then a is also in P ≥ y,z ; so, P ≥ y,z ∩ ( P \ { x } ) = ∅ and this proves that y ∼ z in RU B ( P \ { x } ). In conclusion, RU B ( P ) − x = RU B ( P \ { x } ) and RU B ( P ) ց d RU B ( P \ { x } ). (cid:3) Let us denote by G ◦ the reflexive graph obtained from a graph G by adding loops to its non loopedvertices. We note that, by identifying A ( C ( G )) with V ( G ), we get m ( C ( G )) = G ◦ for every G ∈ G . Theorem 3.3 Let G ∈ G and H a subgraph of G such that C ( G ) ց d C ( H ) . Then G ◦ ց d H ◦ .In particular, if G ∈ G ◦ and H is a subgraph of G , then C ( G ) ց d C ( H ) = ⇒ G ց d H . Proof : By Proposition 3.3, we have two dismantling f G : RU B ( C ( G )) ց d m ( C ( G )) = G ◦ and f H : RU B ( C ( H )) ց d m ( C ( H )) = H ◦ . There is also a dismantling ϕ : RU B ( C ( G )) ց d RU B ( C ( H )) from C ( G ) ց d C ( H ) and Proposition 3.4. So, we have the following diagram: RU B ( C ( G )) m ( C ( G )) = G ◦ RU B ( C ( H )) m ( C ( H )) = H ◦ f G ϕ f H The conclusion G ◦ ց d H ◦ follows from Corollary 2.1 applied to ( G, G ′ , G ′′ ) = ( RU B ( C ( G )) , G ◦ , H ◦ ). (cid:3) ( G , K ) K Let K be the category of finite simplicial complexes (cf. [Koz08] for a reference textbook) and let K ∈ K .If σ is a simplex of K , we write σ ∈ K . A simplicial complex K is a simplicial cone if there is a subcomplex L and a vertex a of K \ L such that the set of simplices of K is {{ a } , σ, { a } ∪ σ, σ ∈ L } ; in this case, K isdenoted aL . Let us recall the following definitions for a vertex x of K : • star oK ( x ) := { σ ∈ K, x ∈ σ }• lk K ( x ) := { σ ∈ K, { x } ∪ σ ∈ K and x / ∈ σ }• star K ( x ) := { σ ∈ K, { x } ∪ σ ∈ K } = star oK ( x ) ∪ lk K ( x ) • K − x := { σ ∈ K, x / ∈ σ } .We note that a simplicial complex K is a simplicial cone if, and only if, one can write K = xL with L = K − x for some vertex x . In [BM09], a notion of dismantlability is defined in the framework of simplicialcomplexes. A vertex x of a simplicial complex K is said dominated by the vertex a of K if lk K ( x ) is asimplicial cone aK ′ for some subcomplex K ′ of K ; in this case, the deletion of the vertex x in K is called an elementary strong collapse and denoted K ցց K − x . A strong collapse, denoted K ցց L , is thesuccession of elementary strong collapses. In this paper, by analogy with the situation in graphs and posets,a dominated vertex in a simplicial complex K will be said dismantlable in K . Remark 4.1 In [CY07], the authors introduce the notion of linear coloring on simplicial complexes. TheTheorem 6.2 of [CY07] shows that the notion of LC-reduction in [CY07, § 6] and the notion of strong reductiondefined in [BM09] are equivalent. ∆ G : G → K Let G ∈ G . We recall that ∆ G ( G ) (sometimes called the clique complex of G ) is the simplicial complexwhose simplices are given by sets of vertices of complete subgraphs of G . The following facts are easy : • If G is a reflexive graph, then G is a cone if, and only if, ∆ G ( G ) is a simplicial cone. • For every vertex x of a graph G , ∆ G ( N G ( x ) − x ) = lk ∆ G ( G ) ( x ).7 emma 4.1 Let G ∈ G , a, x ∈ V ( G ) such that a = x and x looped. Then, x is dominated by a in G if, andonly if, x is dominated by a in ∆ G ( G ) . Proof : Let x a looped vertex. If x ⊢ d a , then N G ( x ) − x is a cone with apex a and lk ∆ G ( G ) ( x ) =∆ G ( N G ( x ) − x ) = aL with L = (cid:0) ∆ G ( N G ( x ) − x ) (cid:1) − a , i.e. x is dominated by a in ∆ G ( G ). Conversely, iflk ∆ G ( G ) ( x ) = ∆ G ( N G ( x ) − x ) is a simplicial cone aL , then necessarily y ∼ a for all y ∈ N G ( x ) − x and x ∼ a ;in other terms, N G ( x ) ⊂ N G ( a ), i.e. x ⊢ d a . (cid:3) Theorem 4.1 Let G, H ∈ G ◦ . Then, G ց d H ⇐⇒ ∆ G ( G ) ցց ∆ G ( H ) . Proof : Follows by iteration of Lemma 4.1. (cid:3) F G : K → G ◦ Let K a simplicial complex. The face graph ([BFJ08]) F G ( K ) of K is the reflexive graph whose vertices arethe non empty simplices of K with an edge between two simplices if one contains the other. If x is a vertexof K , { x } will denote the same vertex as a 0-simplex of K or as a vertex of F G ( K ). More generally, if σ isa simplex of K , we also denote σ the corresponding vertex of F G ( K ). Theorem 4.2 Let K, L ∈ K . Then, K ցց L = ⇒ F G ( K ) ց d F G ( L ) . Proof : It is sufficient to prove K ցց K − x = ⇒ F G ( K ) ց d F G ( K − x ). As V ( F G ( K )) \ V ( F G ( K − x )) = star oK ( x ), we have to verify that one can dismant, one by one, all the elements of star oK ( x ) whenlk K ( x ) is a cone. So, let x a dismantlable vertex in K and a a vertex which dominates x in K ; we havestar oK ( x ) = Γ x ∪ Γ x,a with Γ x := { σ ∈ K, x ∈ σ and a σ } and Γ x,a := { σ ∈ K, x ∈ σ and a ∈ σ } . As theneighborhood in F G ( K ) of a maximal simplex σ of Γ x is {{ a } ∪ σ } ∪ { τ, τ ⊂ σ } , we have σ ⊢ d { a } ∪ σ in F G ( K ). So, all maximal simplices of Γ x are dismantlable and, when they have been deleted, the maximalsimplices of the resulting subset of Γ x are also dismantlable by the same argument and the iteration of thisprocedure showes that all vertices of Γ x are dismantlable (the procedure ends when the 0-simplex { x } isdominated by the 1-simplex { a, x } ). Next, it remains to prove that one can dismant all vertices of Γ x,a . Thisfollows from the existence of a similar procedure to the precedent, in the reverse order. First, the vertex { x, a } is dominated by a . Next, after the removing of { x, a } , vertices of type { a, x, y } are dominated by { a, y } and after the removing of these vertices, vertices of type { a, x, y, z } are dominated by { a, y, z } and soon, until all vertices of Γ x,a have been deleted. (cid:3) Remark 4.2 There is an obvious morphism f ◦ g : F G ( K ) → F G ( K − x ) where g : F G ( K ) → F G ( K ) − Γ x is defined by g ( σ ) = { a } ∪ σ on Γ x and g ( σ ) = σ otherwise and f : F G ( K ) − Γ x → (cid:0) F G ( K ) − Γ x (cid:1) − Γ x,a = F G ( K − x ) is defined by f ( σ ) = σ \ { x } on Γ x,a and f ( σ ) = σ otherwise. Nevertheless, in general, we don’thave g ∼ F G ( K ) , nor f ∼ F G ( K ) − Γ x and the preceding proof shows the necessity of deleting the vertices of star oK ( x ) in a certain order. To establish the reciprocal statement of Theorem 4.2, we need two lemmas. Lemma 4.2 Let K ∈ K and L a subcomplex of K such that F G ( K ) ց d F G ( L ) . If σ is a maximal simplexof K which appears in a dismantling sequence from F G ( K ) to F G ( L ) , then there is a -simplex { x } with x ∈ σ which appears before σ in the same dismantling sequence. Proof : Let us suppose that σ is a maximal simplex of K which appears in a dismantling sequence from F G ( K ) to F G ( L ). This means that after having removed some vertices, we get a subgraph F ′ of F G ( K )and there is a simplex σ ′ which dominates σ in F ′ . As σ is a maximal simplex and σ ∼ σ ′ , we must have σ ′ σ . Now, let x ∈ σ ; σ ⊢ d σ ′ implies { x } ∼ σ ′ , i.e. x ∈ σ ′ . In particular, if no vertex of σ has beendismantled, then σ ⊂ σ ′ . But this contradict σ ′ σ . So, there must be at least one vertex of σ which hasbeen dismantled before σ . (cid:3) Lemma 4.3 Let K ∈ K and L a subcomplex of K such that F G ( K ) ց d F G ( L ) . If { x } is the first 0-simplexdismantled in a dismantling sequence from F G ( K ) to F G ( L ) , then x is dismantlable in K . roof : Let { x } be the first 0-simplex dismantled in a dismantling sequence from F G ( K ) to F G ( L ) and σ a simplex such that { x } ⊢ d σ in the dismantling process. We will show that every element of σ dominates x in K . So, let us take a ∈ σ , a = x and τ ∈ lk K ( x ). We have to prove that τ ∪ { a } is a simplex of lk K ( x ).Let τ max be a maximal simplex of K containing τ ∪ { x } ; by Lemma 4.2, we know that τ max has not beendismantled before x . As x ∈ τ max and { x } ⊢ d σ , we conclude that σ is adjacent to τ max , i.e. σ ⊂ τ max (because τ max is maximal). Consequently, a ∈ τ max and τ ∪ { a, x } ⊂ τ max ; this shows that τ ∪ { a } is a simplex oflk K ( x ). (cid:3) Theorem 4.3 Let K ∈ K and L a subcomplex of K such that F G ( K ) ց d F G ( L ) . Then K ցց L . Proof : By Lemma 4.3, we know that there exists a vertex x of K − L such that K ցց K − x . Now, fromTheorem 4.2, we get a dismantling f x : F G ( K ) ց d F G ( K − x ). So, with the hypothesis of a dismantling ϕ : F G ( K ) → F G ( L ), we have the following triangle: F G ( K ) F G ( K − x ) F G ( L ) f x ϕ which allows to conclude F G ( K − x ) ց d F G ( L ) from Corollary 2.1 because F G ( L ) is a subgraph of F G ( K − x ).Now, we iterate the argument with F G ( K − x ). The iteration ends when all 0-simplices which are not verticesof F G ( L ) have been dismantled and this proves that K ցց L . (cid:3) Remark 4.3 We also deduce from the proof of Theorem 4.3 that a dismantling sequence from K to L isobtained by keeping the 0-simplices (or vertices of K ) in a dismantling sequence from F G ( K ) to F G ( L ) . ( G ◦ , P , K ) The order complex of a poset P ∈ P is the simplicial complex ∆ P ( P ) whose simplices are given by thechains of P . First, we note the elementary facts: • ∆ P ( P >x ∪ P Let P, Q ∈ P . Then, P ց wd Q ⇐⇒ ∆ P ( P ) ցց ∆ P ( Q ) . Remark 5.1 We know from [BM09, Theorem 4.14.a] that P ց d Q implies ∆ P ( P ) ցց ∆ P ( Q ) ; the exampleof the poset P given in Figure 3 ( d is dominated by a in ∆ P ( P ) but not dominated in P ) shows that thereciprocal statement is not true in general. Let K a simplicial complex. The face poset F P ( K ) of K is the poset given by the set of non emptysimplices of K with the inclusion as order relation. From [BM09, Theorem 4.14.b], we know that K ցց L = ⇒ F P ( K ) ց d F P ( L ); the reciprocal statement is true: Theorem 5.2 Let K, L ∈ K . If F P ( K ) ց d F P ( L ) , then K ցց L . Proof : Let us suppose that F P ( K ) ց d F P ( L ) in P . By Proposition 3.2 and identity Comp ◦ F P = F G , F G ( K ) ց d F G ( L ) in G ◦ and, by Theorem 4.3, K ցց L . (cid:3) .2 Homotopy classes Addition or deletion of dismantlable vertices define an equivalence relation in G : [ G ] d = [ H ] d if there is in G a sequence G = J , J , J , . . . , J n − , J n = H such that J i ց d J i +1 or J i d ր J i +1 or J i ∼ = J i +1 ( J i and J i +1 are isomorphic graphs) for i = 0 , , . . . , N − 1. The equivalence class [ G ] d of G will be called the d-homotopytype of G .The term homotopy is given here by analogy with the equivalence class [ P ] d of a poset P in P which isdefined in a similar way by dismantlings in P . It is well known ([Sto66]) that [ P ] d is actually the homotopyclass of the poset P considered as a topological space (with { P ≤ x , x ∈ P } as a base of neighborhoods). In asimilar way and following [BM09, Definition 2.1], two simplicial complexes K and L have the same stronghomotopy type if one can go from K to L by a succession of strong collapses or strong expansions. Remark 5.2 The d -homotopy type is quite rigid. The example (see Figure 4) of the reflexive cycles C ◦ n shows the important gap with the s -homotopy (two graphs G and H have the same s -homotopy type if, andonly if, the simplicial complexes ∆ G ( G ) and ∆ G ( H ) have the same simple homotopy type, cf. [BFJ08]). •• • C ◦ • ••• C ◦ •• • • • C ◦ [ C ◦ n ] d = [ C ◦ m ] d , ∀ n, m ≥ , n = m [ C ◦ n ] s = [ C ◦ m ] s = [ C ◦ ] s , ∀ n, m ≥ Figure 4: d -homotopy classes and s -homotopy classes Proposition 5.1 Let P ∈ P and x a weak dismantlable element in P . Then [ P ] d = [ P \ { x } ] d . Proof : As x is a weak dismantlable element in P , it exists an element a comparable with all elements of P >x ∪ P As a useful consequence of Proposition 5.1, if two posets P and Q are such that P ց wd Q ,then [ P ] d = [ Q ] d . In other terms, the weak dismantlability preserves the homotopy type in P . ( G ◦ , P , K ) G ◦ P K F P ∆ P Comp C ∆ G F G Figure 5: The triangle ( G ◦ , P , K )The functors in the triangle ( G ◦ , P , K ) are compatible with the various homotopy classifications (d-homotopy type in G , homotopy type in P and strong homotopy type in K ):10 heorem 5.3 1. Let G, H ∈ G ◦ .a. G and H have the same d-homotopy type if, and only if, C ( G ) and C ( H ) have the same homotopytype.b. G and H have the same d-homotopy type if, and only if, ∆ G ( G ) and ∆ G ( H ) have the same stronghomotopy type.2. Let P, Q ∈ P .a. P and Q have the same homotopy type if, and only if, Comp ( P ) and Comp ( Q ) have the samed-homotopy type.b. P and Q have the same homotopy type if, and only if, ∆ P ( P ) and ∆ P ( Q ) have the same stronghomotopy type.3. Let K, L ∈ K .a. K and L have the same strong homotopy type if, and only if, F G ( K ) and F G ( L ) have the samed-homotopy type.b. K and L have the same strong homotopy type if, and only if, F P ( K ) and F P ( L ) have the samehomotopy type. Proof : All these equivalences are immediate corollaries of previous results: Theorems 3.2 and 3.3 (1.a),Theorem 4.1(1.b), Theorem 3.1 and Proposition 5.1 (2.a), Theorem 5.1 and Proposition 5.1 (2.b), Theorems4.2 and 4.3 (3.a), [BM09, Theorem 4.14.b] and Theorem 5.2 (3.b). (cid:3) We recall that there is an operation of barycentric subdivision either for graphs, for posets, or for simplicialcomplexes ([BFJ08]) verifying Bd = C ◦ Comp = F G ◦ ∆ G in G ◦ , Bd = Comp ◦ C = F P ◦ ∆ P in P and Bd = ∆ G ◦ F G = ∆ P ◦ F P in K . Proposition 5.2 1. Let G, H ∈ G ; then, G ց d H ⇐⇒ Bd ( G ) ց d Bd ( H ) 2. Let K, L ∈ K ; then, K ցց L ⇐⇒ Bd ( K ) ցց Bd ( L ) 3. Let P, Q ∈ P ; then, P ց wd Q ⇐⇒ Bd ( P ) ց d Bd ( Q ) Proof : The assertions 1 and 2 are corollaries of Theorems 4.1, 4.2 and 4.3 by using, respectively, F G ◦ ∆ G = Bd (in G ) and ∆ G ◦F G = Bd (in K ). The assertion 3 is a consequence of Theorems 3.1, 3.2, 3.3 and equality C ◦ Comp = Bd (in P ). (cid:3) Remark 5.4 If L is reduced to a vertex of K , the assertion 2 of Proposition 5.2 is [BM09, Theorem 4.15]. Hom complex Let G, H ∈ G . The set of morphisms from G to H is the vertex set of the reflexive graph hom G ( G, H ) andis also the set of vertices (or 0-dimensional cells) of the polyhedral complex Hom ( G, H ) ([BK06],[Koz08])whose cells are indexed by functions (which will be called indexing functions ) η : V ( G ) → V ( H ) \ {∅} , suchthat if ( x, y ) ∈ E ( G ), then η ( x ) × η ( y ) ⊂ E ( H ). Example 6.1 We will illustrate the results of this section with the example given by the path G = P (i.e., V ( G ) = { , , } and ∼ ∼ ) and the complete graph H = K (i.e., V ( K ) = { a, b, c } and a ∼ b ∼ c ∼ a ).The notation rst will indicate a morphism from P to K which sends 0 to r , 1 to s and 2 to t . Thereare 12 morphisms from P to K : u v w x y z f g h j k l aca bcb bab cac cbc aba acb bca bac cab abc cba The graph hom G ( P , K ) and the polyhedral complex Hom ( K , P ) are represented in Figure 6. ••••• • • •• •• v gufyk z l jh wx hom G ( P , K ) • ••••• • • •• •• v gufyk z l jh wx Hom ( P , K )Figure 6: The graph hom G ( P , K 3) and the polyhedral complex Hom ( P , K ) Hom ( − , − ) and hom G ( − , − ) For studying the polyhedral complex Hom ( G, H ), it is usual to consider its face poset F P ( Hom ( G, H )) whoseelements are all indexing functions with order given by η ≤ η ′ if and only if η ( x ) ⊂ η ′ ( x ) for all x in V ( G ).Actually, there is a natural identification of F P ( Hom ( G, H )) with a subposet of C ( hom G ( G, H )), theposet of complete subgraphs of hom G ( G, H ). Indeed, let η ∈ F P ( Hom ( G, H )) and for every vertex x of G ,let us choose an element y x ∈ η ( x ). Then the application f : V ( G ) → V ( H ) , x y x is actually a morphismfrom G to H ; such an application will be called an associated morphism to η . The set of all morphismsassociated to η will be called Ψ( η ). By definition of indexing functions, Ψ( η ) induces a complete subgraphof hom G ( G, H ) and we get an injective poset mapΨ : F P ( Hom ( G, H )) −→ C ( hom G ( G, H )) η Ψ( η )which identifies F P ( Hom ( G, H )) with a subposet of C ( hom G ( G, H )).Now let [ f , f , . . . , f k ] ∈ C ( hom G ( G, H )) (i.e., the set { f , f , . . . , f k } of morphisms from G to H induces acomplete subgraph of hom G ( G, H )). We define the indexing function Φ([ f , f , . . . , f k ]) : V ( G ) → V ( H ) \{∅} by Φ([ f , f , . . . , f k ])( x ) = { f ( x ) , f ( x ) , . . . , f k ( x ) } for all x ∈ V ( G ). This gives a morphism of posets:Φ : C ( hom G ( G, H )) ֒ → F P ( Hom ( G, H ))[ f , f , . . . , f k ] Φ([ f , f , . . . , f k ]) Proposition 6.1 Let G, H ∈ G . By identifying F P ( Hom ( G, H )) with a subposet of C ( hom G ( G, H )) , wehave: C ( hom G ( G, H )) ց d F P ( Hom ( G, H )) Proof : First, we note that Φ ◦ Ψ = 1 F P ( Hom ( G,H )) . This implies that (Ψ ◦ Φ) = Ψ ◦ Φ, i.e. Ψ ◦ Φ : C ( Hom ( G, H )) → C ( Hom ( G, H )) is a retraction on F P ( Hom ( G, H )) (identified with Ψ( Hom ( G, H ))). Next,for all [ f , f , . . . , f k ] ∈ C ( hom G ( G, H )), [ f , f , . . . , f k ] ⊂ Ψ ◦ Φ([ f , f , . . . , f k ]), i.e. 1 C ( hom G ( G,H )) ≤ Ψ ◦ Φ.The conclusion follows from Proposition 3.1. (cid:3) Example 6.2 The posets obtained when G = P and K = K are drawed in Figures 7 and 8. The dis-mantling sequence: f uv, guv, f gu, f gv, f g, uv, hwx, jwx, hjw, hjx, wx, hj, kyz, lyz, kly, klz, yz, kl illustratesthe Proposition 6.1. The face graph F G ( Hom ( G, H ) of the polyhedral complex Hom ( G, H ) is the graph whose vertices are theindexing functions of Hom ( G, H ) with edges η ∼ η ′ if and only if either η ( x ) ⊂ η ′ ( x ) for all x in V ( G ), or η ′ ( x ) ⊂ η ( x ) for all x in V ( G ). In other words, F G ( Hom ( G, H )) = Comp ( F P ( Hom ( G, H ))). Corollary 6.1 Let G, H ∈ G . we have the following dismantling in G : Bd ( hom G ( G, H )) ց d F G ( Hom ( G, H )) Proof : Follows from Proposition 6.1 by using Comp ◦ C = Bd , Comp ◦ F P = F G and Proposition 3.2. (cid:3) guv hjwx klyz fuv guv fgu fgv hwx jwx hjw hjx klzklylyzkyzuv fu gu fg fv gv vw wx hw jw hj hx jx xy lzkzkllykyyz uz u f g v w h zlkyxj Figure 7: C ( hom G ( P , K • a, bca, b • b, cab, c • a, cba, c • • • • aca, b • • • • • • bab, cba, cb ca, bc • • • • a, cbc • ab, ca • • • • • • •••••• u f g v w h zlkyxjFigure 8: F P ( Hom ( P , K Hom ( G, H ) and foldings in G or in H Theorem 6.1 Let G, H ∈ G .1. If a is dismantlable in G , then F G ( Hom ( G, H )) ց d F G ( Hom ( G − a, H )) (by identifying F G ( Hom ( G − a, H )) with a subgraph of F G ( Hom ( G, H )) ).2. If u is dismantlable in H , then F G ( Hom ( G, H )) ց d F G ( Hom ( G, H − u )) (by identifying F G ( Hom ( G, H − u )) with a subgraph of F G ( Hom ( G, H )) ). Proof : 1. We have the following diagram where the morphisms A and A ′ are dismantlings given byCorollary 6.1 and the morphism B is a dismantling given by Propositions 2.1 and 5.2.1: Bd ( hom G ( G, H )) F G ( Hom ( G, H )) Bd ( hom G ( G − a, H )) F G ( Hom ( G − a, H )) AB A ′ By considering ( G, G ′ , G ′′ ) = ( Bd ( hom G ( G, H )) , F G ( Hom ( G, H )) , F G ( Hom ( G − a, H ))), the conclusion fol-lows from Corollary 2.1.2. The proof is similar. (cid:3) Example 6.3 Returning to the case G = P and H = K , we have ⊢ d in P and P ց d P − K .The deletion in F G ( Hom ( P , K )) of the twelve numbered vertices in the order indicated in Figure 8 followedby the deletion of the vertices f , g , h , j , k and l is a dismantling sequence from F G ( Hom ( P , K )) to G ( Hom ( K , K )) (identified with a subgraph of F G ( Hom ( P , K )) ). We note that F G ( Hom ( K , K )) is astiff graph. • ••••• • • •• •• •• • •• • ••• • ••• • ••• • 57 68 23 4111109 12 v gufyk z l jh wx ••• • •• •• • •• • v ′ u ′ y ′ z ′ w ′ x ′ u ′ v ′ w ′ a c b c b ax ′ y ′ z ′ c a c b a b Figure 9: The graphs F G ( Hom ( P , K F G ( Hom ( K , K G ◦ F G = Bd (in K ) and Theorem 4.1, we get the following corollary: Corollary 6.2 Let G, H ∈ G .1. If a is dismantlable in G , then Bd ( Hom ( G, H )) ցց Bd ( Hom ( G − a, H )) In particular, Bd ( Hom ( G, H )) and Bd ( Hom ( G − a, H )) have the same strong homotopy type.2. 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