Higher Independence Complexes of graphs and their homotopy types
aa r X i v : . [ m a t h . A T ] J a n HIGHER INDEPENDENCE COMPLEXES OF GRAPHS AND THEIRHOMOTOPY TYPES
PRIYAVRAT DESHPANDE AND ANURAG SINGH
Abstract.
For r ≥
1, the r -independence complex of a graph G is a simplicial complex whosefaces are subset I ⊆ V ( G ) such that each component of the induced subgraph G [ I ] has at most r vertices. In this article, we determine the homotopy type of r -independence complexes ofcertain families of graphs including complete s -partite graphs, fully whiskered graphs, cyclegraphs and perfect m -ary trees. In each case, these complexes are either homotopic to a wedgeof equi-dimensional spheres or are contractible. We also give a closed form formula for theirhomotopy types. Introduction
Let G be a simple undirected graph. A subset I ⊆ V ( G ) of vertex set of G, is called an independent set if the vertices of I are pairwise non-adjacent in G . The independence complex of G , denoted Ind ( G ), is a simplicial complex whose faces are the independent subsets of V .The study of homotopy type of independence complexes of graphs has received a lot of attentionin last two decades. For example, in Babson and Kozlov’s proof of Lov´asz’s conjecture (in [1])regarding odd cycles and graph homomorphism complexes the independence complexes of cyclegraphs played an important role. In [17] , Meshulam related homology groups of Ind ( G )with the domination number of G . The problem of determining a closed form formula for thehomotopy type of Ind ( G ) for various classes of graphs is also well studied. For instance, see[16] for paths and cycle graphs, [13] for forests, [4, 5] for grid graphs, [14] for chordal graphsand [10] for categorical product of complete graphs and generalized mycielskian of completegraphs. Barmak [2] studied the topology of independence complexes of triangle-free graphs andclaw-free graphs. He also gave a lower bound for the chromatic number of G in terms of thestrong Lusternik-Schnirelmann category of Ind ( G ).Recently in [19], Paolini and Salvetti generalized the notion of independence complexes bydefining r -independence complex for any r ≥
1. For a graph G , a subset I ⊆ V ( G ) is called r -independent if each connected component of the induced subgraph G [ I ] has at most r vertices.For r ≥
1, the r -independence complex of G , denoted Ind r ( G ) is a simplicial complex whosefaces are all r -independent subsets of V ( G ). They established a relationship between the twistedhomology of the classical braid groups and the homology of higher independence complexes ofassociated Coxeter graphs. In particular they showed that r -independence complexes of pathgraphs are homotopy equivalent to a wedge of spheres (see Theorem 4.2).The aim of this article is to initiate the study of these so-called higher independence complexes of graphs. Our focus is on determining a closed form formula for its homtopy type. In the articlewe identify several classes of graphs for which these complexes are either homotopic to a wedgeof equi-dimensional spheres or are contractible. In each case we also determine the dimensionof the spheres and their number; we achieve this using discrete Morse theory.The paper is organized as follows. In Section 2 we recall all the important definitions andrelevant tools from discrete Morse theory. The formal definition and basic properties of higherindependence complexes is given in Section 3; here we also look at the complexes associated Mathematics Subject Classification.
Key words and phrases.
Independence complex, higher independence complex, fully whiskered graphs, cyclegraphs, perfect binary trees.PD and AS are partially funded by a grant from Infosys Foundation. PD is also partially funded by theMATRICS grant MTR/2017/000239. ith complete s -partite graphs and show that they are always homtopic to a wedge of spheres.We also show that if a graph is modified by attaching leaves to every vertex then the higherindependence complexes of these new graphs are either wedge of spheres or are contractible.In Section 4 we consider the case of cycle graphs and in Section 5 we consider perfect m -arytrees; in both the cases the associated complexes are either wedge of spheres or are contractible.Moreover, in both the cases we construct optimal discrete Morse functions on these complexes.As a result all the critical cells are concentrated in a fixed dimension. The construction of theseMorse functions as well as the formula for the number of critical cells both are combinatoriallyinvolved. Finally in Section 6 we outline some questions and conjectures.2. Preliminaries
Let G be a simple, undirected graph and v ∈ V ( G ) be a vertex of G . The total number ofvertices adjacent to v is called degree of v , denoted deg( v ). If deg( v ) = 1, then v is called a leaf vertex. A graph H with V ( H ) ⊆ V ( G ) and E ( H ) ⊆ E ( G ) is called a subgraph of the graph G . For a nonempty subset U of V ( G ), the induced subgraph G [ U ], is the subgraph of G withvertices V ( G [ U ]) = U and E ( G [ U ]) = { ( a, b ) ∈ E ( G ) : a, b ∈ U } . In this article, G [ V ( G ) \ A ]will be denoted by G − A for A ( V ( G ). Definition 2.1.
An (abstract) simplicial complex K on a finite set X is a collection of subsetssuch that ( i ) ∅ ∈ K , and ( ii ) if σ ∈ K and τ ⊆ σ , then τ ∈ K . The elements of K are called simplices of K . If σ ∈ K and | σ | = k + 1, then σ is saidto be k -dimensional (here, | σ | denotes the cardinality of σ as a set). Further, if σ ∈ K and τ ⊆ σ then τ is called a face of σ and if τ = σ then τ is called a proper face of σ . The setof 0-dimensional simplices of K is denoted by V ( K ), and its elements are called vertices of K .A subcomplex of a simplicial complex K is a simplicial complex whose simplices are containedin K . For s ≥
0, the k -skeleton of a simplicial complex K , denoted K ( s ) , is the collection ofall those simplices of K whose dimension is at most s . In this article, we do not distinguishbetween an abstract simplicial complex and its geometric realization. Therefore, a simplicialcomplex will be considered as a topological space, whenever needed.Let S r denotes a sphere of dimension r and ∗ denotes join of two spaces. The following resultswill be used repeatedly in this article. Lemma 2.2 ([3, Lemma 2.5]) . Suppose that K and K are two finite simplicial complexes. (1) If K and K both have the homotopy type of a wedge of spheres, then so does K ∗ K . (2) (cid:16) W i S a i (cid:17) ∗ (cid:16) W j S b j (cid:17) ≃ W i,j S a i + b j +1 We now discuss some tools needed from discrete Morse theory. The classical reference forthis is [9]. However, here we closely follow [15] for notations and definitions.
Definition 2.3 ([15, Definition 11.1]) . A partial matching on a poset P is a subset M ⊆ P × P such that (i) ( a, b ) ∈ M implies a ≺ b ; i.e., a < b and no c satisfies a < c < b , and (ii) each a ∈ P belong to at most one element in M . Note that, M is a partial matching on a poset P if and only if there exists A ⊂ P and aninjective map µ : A → P \ A such that µ ( a ) ≻ a for all a ∈ A .An acyclic matching is a partial matching M on the poset P such that there does not exista cycle µ ( a ) ≻ a ≺ µ ( a ) ≻ a ≺ µ ( a ) ≻ a . . . µ ( a t ) ≻ a t ≺ µ ( a ) , t ≥ . For an acyclic partial matching on P , those elements of P which do not belong to the matchingare called critical . he main result of discrete Morse theory is the following. Theorem 2.4 ([15, Theorem 11.13]) . Let K be a simplicial complex and M be an acyclicmatching on the face poset of K . Let c i denote the number of critical i -dimensional cells of K with respect to the matching M . Then K is homotopy equivalent to a cell complex K c with c i cells of dimension i for each i ≥ , plus a single -dimensional cell in the case where the emptyset is also paired in the matching. Following can be inferred from Theorem 2.4.
Corollary 2.5.
If an acyclic matching has critical cells only in a fixed dimension i , then K ishomotopy equivalent to a wedge of i -dimensional spheres. Corollary 2.6.
If the critical cells of an acyclic matching on K form a subcomplex K ′ of K ,then K simplicially collapses to K ′ , implying that K ′ is homotopy equivalent to K . In this article, by matching on a simplicial complex K , we will mean that the matching ison the face poset of K . Let K be a simplicial complex with vertex set X and N x = { σ ∈ K : σ \ { x } , σ ∪ { x } ∈ K } be a subcomplex of K , where x ∈ X . Define a matching on K using x as follows: M x = { ( σ \ { x } , σ ∪ { x } ) : σ \ { x } , σ ∪ { x } ∈ K } . Definition 2.7.
Matching M x , as defined above, is called an element matching on K usingvertex x . The following result tells us that an element matching is always acyclic.
Lemma 2.8 ([18, Lemma 3.2]) . The matching M x is an acyclic matching on K and perfectacyclic matching on N x . To obtain an acyclic matching on a simplicial complex K , the next result tells us that onecan define a sequence of element matchings on K using its vertices. Proposition 2.9 ([10, Proposiotion 3.1]) . Let K be a simplicial complex and x , x , . . . , x n arevertices of K . Then, n F i =1 M x i is an acyclic matching on K , where M x i = { ( σ \{ x i } , σ ∪ { x i } ) : σ \ { x i } , σ ∪ { x i } ∈ K i } and K i +1 = K i \ { σ : σ ∈ η for some η ∈ M x i } for i ∈ { , . . . , n } . Proposition 2.9 will be used heavily in this article. Another useful way to construct an acyclicmatching on a poset P is to first map P to some other poset Q , then construct acyclic matchingson the fibers of this map and patch these acyclic matchings together to form an acyclic matchingfor the whole poset. Theorem 2.10 (Patchwork theorem [15, Theorem 11.10]) . If ϕ : P → Q is an order-preservingmap and for each q ∈ Q , the subposet ϕ − ( q ) carries an acyclic matching M q , then F q ∈ Q M q isan acyclic matching on P. The following result is a special case of Theorem 2.10.
Theorem 2.11 ([11, Lemma 4.3]) . Let K and K be disjoint families of subsetes of a finiteset such that τ * σ if σ ∈ K and τ ∈ K . If M i is an acyclic matching on K i for i = 0 , then M ∪ M is an acyclic matching on K ∪ K . Basic results for higher independence complex
We begin this section by exploring some basic results related to the main object of this article, i.e. , higher independence complex. Henceforth, unless otherwise mentioned, r ≥ n ] will denote the set { , . . . , n } . Definition 3.1.
Let G be a graph and A ⊆ V ( G ) . Then A is called r -independent if connectedcomponents of G [ A ] have cardinality at most r . efinition 3.2. Let G be a graph and r ∈ N . The r -independence complex of G , denoted Ind r ( G ) has vertex set V ( G ) and its simplices are all r -independent subsets of V ( G ) . Example 3.3.
Fig. 1 shows a graph G , its -independence complex and -independence complex.The -independence complex of G consists of maximal simplices, namely { v , v , v } and { v } .The complex Ind ( G ) consists of maximal simplices, namely { v , v } , { v , v } , { v , v } and { v , v , v } . v v v v (a) G v v v v (b) Ind ( G ) v v v v (c) Ind ( G ) Figure 1
The following are some easy observations from the definition of r -independence complex. Observation 3.4. ( i ) For any graph G , Ind r ( G ) is ( r − -connected. Moreover, if r ≥| V ( G ) | then Ind r ( G ) ≃ { point } . ( ii ) If G is connected graph and | V ( G ) | = r + 1 , then Ind r ( G ) ≃ S r − . ( iii ) Let K n be the complete graph on n vertices, then Ind r ( K n ) is equal to ( r − th skeletonof an ( n − -simplex, denoted ∆ n − , i.e.,Ind r ( K n ) = (∆ n − ) ( r − . ( iv ) If G and H are two disjoint graphs, then Ind r ( G ⊔ H ) ≃ Ind r ( G ) ∗ Ind r ( H ) . ( v ) If G has a non-empty connected component of cardinality at most r , then Ind r ( G ) iscontractible. In Observation 3.4( iii ), we saw that Ind r ( K n ) is homotopic equivalent to a wedge of spheresof dimension r −
1. So one would expect a similar result for complete s -partite graphs for s ≥ complete s -partite graph is a graph in which vertex set can be decomposed into s disjoint sets V , V , . . . , V s such that no two vertices within the same set V i are adjacent and if v ∈ V i and w ∈ V j for i = j then v is adjacent to w . Theorem 3.5.
Let s ≥ and r ≥ . Given m , m , . . . , m s ≥ , the homotopy type of r th independence complex of the complete s -partite graph K m ,...,m s is given as follows, Ind r ( K m ,...,m s ) ≃ _ t S r − , where t = (cid:18) M − r (cid:19) − s X i =1 (cid:18) m i − r (cid:19) and M := P si =1 m i Proof.
For simplicity of notations, we denote K m ,...,m s by G in this proof. Let V , V , . . . , V s bethe partition of vertices of G and V i = { v i , . . . , v m i i } for i ∈ [ s ]. We now define a sequence ofelement matching on ∆ := Ind r ( G ) using vertices v , v , . . . , v s . For i ∈ [ s ], define M i = { ( σ, σ ∪ v i ) : v i / ∈ σ and σ, σ ∪ v i ∈ ∆ i − } ,N i = { σ ∈ ∆ i − : σ ∈ η for some η ∈ M i } , and∆ i = ∆ i − \ N i . sing Proposition 2.9, we get that M = s F i =1 M i is an acyclic matching on Ind r ( G ) with ∆ s asthe set of the critical cells. Claim 1.
The set of critical cells after s th element matching is given as follows: ∆ s = { σ ∈ Ind r ( G ) : | σ | = r, v i / ∈ σ ∀ i ∈ [ s ] and σ * V i for any i ∈ [ s ] } G { σ ∈ Ind r ( G ) : | σ | = r, v / ∈ σ and v i ∈ σ for some i ∈ { , . . . , s }} . Proof of Claim 1.
Clearly, if | σ | = r, v i / ∈ σ ∀ i ∈ [ s ] and σ * V i for any i ∈ [ s ] then G [ σ ∪ v i ]is a connected graph of cardinality r + 1 implying that σ / ∈ N i for all i ∈ [ s ]. Therefore, { σ ∈ Ind r ( G ) : | σ | = r, v i / ∈ σ ∀ i ∈ [ s ] and σ * V i for any i ∈ [ s ] } ⊆ ∆ s . Now, let | σ | = r and v i ∈ σ for some i ∈ { , . . . , s } . For i ∈ { , . . . , s } , if v i ∈ σ then σ \ v i ∈ N implies that σ / ∈ N i . If v j / ∈ σ , then | σ | = r and v i ∈ σ for some i = j implies that G [ σ ∪ v j ] is connectedsubgraph of cardinality r + 1, hence σ / ∈ N j . Thus { σ ∈ Ind r ( G ) : | σ | = r, v / ∈ σ and v i ∈ σ for some i ∈ { , . . . , s }} ⊆ ∆ s .Now consider σ ∈ ∆ s . If σ ⊆ V or | σ | < r or v ∈ σ , then σ ∈ N . If σ ⊆ V i for some i ∈ [ s ]and v i / ∈ σ . Then σ ∪ v i ∈ Ind r ( G ) implying that σ ∈ N i which is a contradiction to the factthat σ ∈ ∆ s . Thus, either σ * V i for any i ∈ [ s ] or if σ ⊆ V i for some i ∈ { , . . . , s } then v i ∈ σ .Now, let | σ | > r . σ ∈ Ind r ( G ) implies that σ ⊆ V i for some i ∈ [ s ] but then σ ∈ N i . Therefore, σ = r . This completes the proof of Claim 1. (cid:3) Using Claim 1, we get that M is an acyclic matching on Ind r ( G ) with exactly | ∆ s | criticalcells of dimension ( r − r ( G ) is homotopy equivalentto a wedge of | ∆ s | spheres of dimension r −
1. We now compute the cardinality of the set ∆ s .Using Claim 1, we get | ∆ s | = (cid:18) s P i =1 m i − sr (cid:19) − s X i =1 (cid:18) m i − r (cid:19) + s X j =2 (cid:18) s P i =1 m i − jr − (cid:19) = (cid:18) s P i =1 m i − r (cid:19) − s X i =1 (cid:18) m i − r (cid:19) This completes the proof of Theorem 3.5. (cid:3)
We now show that adding a whisker (a leaf vertex) at each vertex of G simplifies the homotopytype of higher independence complex. By adding a whisker at vertex v of G , we mean a newvertex is attached to v (the induced subgraph K is called whisker ). We show that the higherindependence complex of fully whiskered graphs is homotopy equivalent to a wedge of equi-dimensional spheres. Definition 3.6.
Given a graph G , a fully whiskered graph of G , denoted W ( G ) , is a graph inwhich a whisker is added to each vertex of G . a a a (a) P a a a a , a , a , (b) W ( P ) Figure 2 heorem 3.7. Let G be a connected graph and V ( G ) = { a , a , . . . , a n } be the set of verticesof G . The homotopy type of Ind r ( W ( G )) is given by the following formula: Ind r ( W ( G )) ≃ W ( n − r − n ) S r − , if n ≤ r ≤ n − , { point } , otherwise . Proof.
Let { b , b , . . . , b n } denote the set of leaves of graph W ( G ) such that b i is adjacent to a i for each i ∈ [ n ]. Let ∆ = Ind r ( W ( G )). We define a sequence of element matching on ∆ usingthe leaf vertices. For i ∈ [ n ], define M ( b i ) = { ( σ, σ ∪ b i ) : b i / ∈ σ, and σ, σ ∪ b i ∈ ∆ i − } ,N ( b i ) = { σ ∈ ∆ i − : σ ∈ m for some m ∈ M ( b i ) } and∆ i = ∆ i − \ N ( b i ) . (1) Claim 2. If σ ∈ Ind r ( W ( G )) and V ( G ) * σ then σ / ∈ ∆ n , i.e. σ is not a critical cell. Let p = min { i : a i / ∈ σ } . From Eq. (1), σ belongs to N ( b p ), which implies that σ / ∈ ∆ n . Thisprove Claim 2.Firstly, let r < n . Since G is connected, if σ ∈ Ind r ( W ( G )) then V ( G ) * σ . Hence, resultfollows from Claim 2 and Corollary 2.5.Secondly, assume that r ≥ n . From definition of Ind r ( G ), it is easy to see that if σ ∈ Ind r ( G )and cardinality of σ is less than r then σ ∈ N ( b ). Thus, if σ ∈ ∆ n then cardinality of σ is atleast r and b / ∈ σ . Using Claim 2, we see that if σ ∈ ∆ n then V ( G ) ⊆ σ . Further, if σ ∈ Ind r ( G )and V ( G ) ⊆ σ then σ / ∈ N ( b i ) for any i ∈ [ n ]. Which shows that σ ∈ ∆ n iff V ( G ) ⊆ σ , a / ∈ σ and | σ | ≥ r . Moreover, V ( G ) ⊆ σ implies that G [ σ ] is always connected. Therefore, cardinalityof σ is exactly r . Combining all these arguments together, we see that ∆ n is a set of (cid:18) n − r − n (cid:19) cells of dimension r −
1. Thus the result follows from Corollary 2.5. (cid:3)
We now show that, for a graph G , adding more whiskers at non-leaf vertices of W ( G ) doesnot affect the connectivity of the higher independence complex. In particular, we give closedform formula for the homotopy type of r -independence complexes of these new graphs. Theorem 3.8.
Let G be a connected graph and W = { a , a , . . . , a n } be the set of all non-leafvertices of G . For i ∈ { , . . . , n } , let l i denote the number of leaves adjacent to vertex a i . If l i > for all i ∈ { , . . . , n } , then the homotopy type of Ind r ( G ) is given as follows. Ind r ( G ) ≃ W t S r − , if r ≥ n, { point } , otherwise , where t = (cid:18)P ni =1 l i − r − n (cid:19) .Proof. Arguments in this proof are similar to that of in proof of Theorem 3.7. For i ∈ [ n ],let { b i, , b i, , . . . , b i,l i } denote the set of leaves adjacent to a i . Let ∆ = Ind r ( G ). We define asequence of element matching on ∆ using leaf vertices b , , b , , . . . , b n, . For i ∈ [ n ], define M ( b i, ) = { ( σ, σ ∪ b i, ) : b i, / ∈ σ, and σ, σ ∪ b i, ∈ ∆ i − } ,N ( b i, ) = { σ ∈ ∆ i − : σ ∈ m for some m ∈ M ( b i, ) } and∆ i = ∆ i − \ N ( b i, ) . (2) Claim 3. If σ ∈ Ind r ( G ) and W * σ then σ / ∈ ∆ n , i.e. σ is not a critical cell. Let p = min { i : a i / ∈ σ } . From Eq. (2), σ belongs to N ( b p, ), which implies that σ / ∈ ∆ n .This prove Claim 3. irstly, let r < n . Since G is connected and W is collection of all non-leaf vertices, G [ W ]is connected subgraph of cardinality n . Therefore, if σ ∈ Ind r ( G ) then W * σ . Hence, resultfollows from Claim 3 and Corollary 2.5.Secondly, assume that r ≥ n . From definition of Ind r ( G ), it is easy to see that if σ ∈ Ind r ( G )and cardinality of σ is less than r then σ ∈ N ( b , ). Thus, if σ ∈ ∆ n then cardinality of σ is atleast r and b , / ∈ σ . Using Claim 3, we see that if σ ∈ ∆ n then W ⊆ σ . Further, if σ ∈ Ind r ( G )and W ⊆ σ then σ / ∈ N ( b i, ) for any i ∈ [ n ]. Which shows that σ ∈ ∆ n iff W ⊆ σ , b , / ∈ σ and | σ | ≥ r . Moreover, W ⊆ σ implies that G [ σ ] is always connected. Therefore, cardinality of σ isexactly r . Combining all these arguments together, we see that ∆ n is a set of (cid:18)P ni =1 l i − r − n (cid:19) cells of dimension r −
1. Thus the result follows from Corollary 2.5. (cid:3)
For n ≥
1, a path graph of length n , denoted P n , is a graph with vertex set V ( P n ) = { , . . . , n } and edge set E ( P n ) = { ( i, i +1) | ≤ i ≤ n − } . For n ≥
3, a cycle graph , denoted C n , is a graphwith vertex set V ( C n ) = { , . . . , n } and edge set E ( C n ) = { ( i, i + 1) | ≤ i ≤ n − } ∪ { (1 , n ) } .We can now compute r -independence complexes of almost all caterpillar graphs. A caterpillar graph is a path graph with some whiskers on vertices. Definition 3.9.
Let G be a graph with V ( G ) = { a , . . . , a n } and L = { l , . . . , l n } be a set of n non-negative integers. Define a graph G L with the following data: V ( G L ) = V ( G ) ⊔ G l i > { b i, , . . . , b i,l i } E ( G L ) = E ( G ) ⊔ G l i > { ( a i , b i,j ) : 1 ≤ j ≤ l i } See Fig. 3 for examples. Clearly, P Ln is a caterpillar graph. a a a a , a , a , a , (a) P (2 , , b b b b , b , b , b , (b) C (2 , , Figure 3
Corollary 3.10.
Given L = ( l , l , . . . , l n ) with l i > for every i ∈ { , , . . . , n } . Then, Ind r ( P Ln ) ≃ Ind r ( C Ln ) ≃ W ( P ni =1 li − r − n ) S r − , if r ≥ n, { point } , otherwise . Higher Independence Complexes of cycle graphs
Kozlov, in [15], computed the homotopy type of 1-independence complex of cycle graphsusing discrete Morse theory. He proved the following result:
Proposition 4.1 ([15, Proposition 11.17]) . For any n ≥ , we have Ind ( C n ) ≃ ( S k − W S k − , if n = 3 k,S k − , if n = 3 k ± . n this section, we generalize this result and compute the homotopy type of Ind r ( C n ) for any n ≥ r ≥
1. In particular, we define a perfect acyclic matching on Ind d − ( C n ). We willuse the following result, proved by Paolini and Salvetti in [19]. Theorem 4.2 ([19, Proposition 3.7]) . For d ≥ , we have Ind d − ( P n ) ∼ = ( S dk − k − , if n = dk or n = dk − { point } , otherwise . To make our computations of Ind d − ( C n ) easier, we first improve the acyclic matching definedby Paolini and Salvetti on Ind r ( P n ), and get a perfect acyclic matching on Ind d − ( P n ). Proposition 4.3.
There exists a perfect acyclic matching on
Ind d − ( P n ) . In particular, if n = dk or dk − and { , , . . . , n } is the vertex set of P n , then the only critical cell is k − F i =0 { di +2 , . . . , di + d − } .Proof. Let n = dk − t for some t ∈ { , , . . . , d − , } , let ∆ = { σ ∈ Ind d − ( P n ) : σ ∩{ d, d, . . . , dk } 6 = ∅} and let ∆ = Ind d − ( P n ) \ ∆. In [19, Proposition 3.7], Paolini and Salevtticonstructed an acyclic matching M on Ind d − ( P n ) with ∆ as the set of critical cells. Here, weconstruct an acyclic matching on ∆ . For i ∈ { , . . . , k − } , define M i = { ( σ, σ ∪ { di + 1 } ) : di + 1 / ∈ σ and σ, σ ∪ di + 1 ∈ ∆ i } ,N i = { σ ∈ ∆ i : σ ∈ η for some η ∈ M i } , and∆ i +1 = ∆ i \ N i . From Proposition 2.9, M ′ = k − F i =0 M i is an acyclic matching on ∆ with ∆ k as the set of criticalcells. Clearly, if n = dk or dk − k = { σ } , where σ = k − F i =0 { di + 2 , . . . , di + d − } .Further, if n = dk, dk − N k − = ∆ k − . Using Theorem 2.11, we get that M ⊔ M ′ is anacyclic matching on Ind d − ( P n ) with ∆ k as set of critical cells. This completes the proof ofProposition 4.3. (cid:3) Following are some immediate corollaries of Proposition 4.3.
Corollary 4.4.
Let d ≥ and G be disjoint union of m path graphs of lengths d or d − . Thenthere exists an acyclic matching on Ind d − ( G ) with exactly one critical cell of dimension andone of dimension ( d − m + m − dm − m − . Corollary 4.5.
Let d ≥ and G be disjoint union of m path graphs. If any connected componentof G has length less than d − or greater than d and less than d − , then there exists an acyclicmatching on Ind d − ( G ) with no critical cell. From Observation 3.4( i ) and ( ii ), we get that Ind d − ( C n ) ≃ { point } for all n ≤ d − d − ( C d − ) ≃ S d − . We now determine the homtopy type of Ind d − ( C n ) for n ≥ d . The ideaof this proof is to define acyclic matching of subsets of face poset of Ind r ( C n ) and then useTheorem 2.10. Theorem 4.6.
For n ≥ d ≥ , we have Ind d − ( C n ) ∼ = _ d − S dk − k − , if n = dk ; S dk − k − , if n = dk + 1; S dk − k , if n = dk + 2; ... ... S dk − k + d − , if n = dk + ( d − . roof. In this proof, we assume that the vertices of C n are labeled as 1 , , . . . , n anti-clockwise.Let k denote the maximal integer such that dk ≤ n . Furthermore, let E be a chain with k + 1elements labeled as follows: e d > e d > · · · > e dk > e r . We define a map(3) φ : F (Ind d − ( C n )) → E by the following rule. The simplices that contain the vertex labeled d get mapped to e d ; thesimplices that do not contain the vertex labeled d , but contain the vertex labeled 2 d get mappedto e d ; the simplices that do not contain the vertices labeled d and 2 d , but contain the vertexlabeled 3 d get mapped to e d ; and so on. Finally, the simplices that does not contain any ofthe vertices labeled d, d, . . . , dk all get mapped to e r .Clearly, the map φ is order-preserving, since if one takes a larger simplex, it will have morevertices, and the only way its image may change is to go up when a new element from the set { d, d, . . . , dk } is added and is smaller than the previously smallest one.Let us now define acyclic matchings on the preimages of elements of E under the map φ . Wesplit our argument into cases. Case 1:
We first consider the preimages φ − ( e d ) through φ − ( e dk ). Let t be an integer suchthat 2 ≤ t ≤ k . The preimage φ − ( e dt ) consists of all simplices σ such that d, d, . . . , d ( t − σ ,while dt ∈ σ . Since σ ∈ Ind d − ( C n ), { dt − , dt − , . . . , dt − ( d − } * σ . This means thatthe pairing σ ↔ σ ∪ { dt − ( d − } provides a well-defined matching, which is acyclic fromLemma 2.8. Case 2:
Next, we consider the preimage φ − ( e d ). For σ ∈ Ind d − ( C n ), let conn d ( σ ) is thenumber of vertices of connected component of C n [ σ ] containing vertex labeled d . We define amap ψ : φ − ( e d ) → { c < c < · · · < c d − } ψ ( σ ) = c , if conn d ( G [ σ ]) is 1 ,c , if conn d ( G [ σ ]) is 2 , ... c d − , if conn d ( G [ σ ]) is d − . Clearly, ψ is a poset map and for i ∈ { , . . . , d − } , if σ ∈ ψ − ( c i ) then cardinality of σ is atleast i .For t ≥
1, let P { i +1 ,...,i + t } t denote the path graph of length t whose vertices are labeled as i + 1 , i + 2 , . . . , i + t (see Fig. 4). i + 1 i + 2 i + t − i + t − i + t Figure 4. P { i +1 ,...,i + t } t We now define a matching on φ − ( e d ) if d − Step : For p ≥
1, it is clear that the p -cells of ψ − ( c ) are in 1-1 correspondence with the p − d − ( P { d +2 ,...,n, ,...,d − } n − ) with one extra cell of dimension 0, which is { d } . UsingProposition 4.3, let M be a perfect matching on Ind d − ( P { d +2 ,...,n, ,...,d − } n − ). Define a matching M on ψ − ( c ) as follows: ( σ, τ ) ∈ M iff ( σ ∪ d, τ ∪ d ) ∈ M . Therefore, we get the following. • Matching M is an acyclic matching on ψ − ( c ) with the following property. If n − dk − n − dk , i.e., n = dk + 2 or dk + 3, then there is only one critical cell of imension dk − k and that is { d } ⊔ k − G i =1 { id + 3 , . . . , ( i + 1) d } ⊔ { , . . . , d − } , if n = dk + 2 , { d } ⊔ k − G i =1 { id + 3 , . . . , ( i + 1) d } ⊔ { n, , . . . , d − } , if n = dk + 3 . (4) Otherwise, there is no critical cell. Step : Observe that, in C n , there are exactly two connected subgraphs of cardinality twocontaining vertex d , which are C n [ { d − , d } ] = P { d − ,d } and C n [ { d, d + 1 } ] = P { d,d +1 } . Thus,cells of ψ − ( c ) can be partitioned into two smaller disjoint subsets ∆ { d − ,d } and ∆ { d,d +1 } .Here, ∆ { d − ,d } is collection of all those cells σ ∈ ψ − ( c ) such that { d − , d } is the connectedcomponent of C n [ σ ]. Similarly, ∆ { d,d +1 } is collection of all those cells σ ∈ ψ − ( c ) such that { d, d + 1 } is the connected component of C n [ σ ]. Clearly, ψ − ( c ) = ∆ { d − ,d } ∪ ∆ { d,d +1 } and∆ { d − ,d } ∩ ∆ { d,d +1 } = ∅ . Now, the idea is to define acyclic matching on ∆ { d − ,d } , ∆ { d,d +1 } andmerge them together to get an acyclic matching on ψ − ( c ).(1) Observe that, for p ≥
2, the p -cells of ∆ { d − ,d } are in 1-1 correspondence with the p − d − ( P { d +2 ,...,n, ,...,d − } n − ) with one extra cell of dimension 1, which is { d − , d } .Using Proposition 4.3, let M be a perfect matching on Ind d − ( P { d +2 ,...,n, ,...,d − } n − ). Definea matching M on ∆ { d − ,d } as follows: ( σ, τ ) ∈ M iff ( σ ∪ { d − , d } , τ ∪ { d − , d } ) ∈ M .Therefore, we get the following.Matching M is an acyclic matching on ∆ { d − ,d } with the following property. If n − dk − dk , i.e., n = dk + 3 or dk + 4, then there is only one critical cell ofdimension dk − k + 1 and that is { d − , d } ⊔ k − G i =1 { id + 3 , . . . , ( i + 1) d } ⊔ { n, , . . . , d − } , if n = dk + 3 , { d − , d } ⊔ k − G i =1 { id + 3 , . . . , ( i + 1) d } ⊔ { n − , n, , . . . , d − } , if n = dk + 4 . (5) Otherwise, there is no critical cell.(2) Similar to the case of ∆ { d − ,d } and using the matching of Ind d − ( P { d +3 ,...,n, ,...,d − } n − ), weget an acyclic matching, say M on ∆ { d,d +1 } with the following property.If n − dk − dk , i.e., n = dk + 3 or dk + 4, then there is only one critical cellof dimension dk − k + 1 and that is { d, d + 1 } ⊔ k − G i =1 { id + 4 , . . . , ( i + 1) d + 1 } ⊔ { , . . . , d − } , if n = dk + 3 , { d, d + 1 } ⊔ k − G i =1 { id + 4 , . . . , ( i + 1) d + 1 } ⊔ { n, , . . . , d − } , if n = dk + 4 . (6) Otherwise, there is no critical cell.Since ψ − ( c ) = ∆ { d − ,d } ⊔ ∆ { d,d +1 } , M = M ⊔ M (defined above) is an acyclic matching on ψ − ( c ) with exactly two critical cells of dimension dk − k + 1 whenever n = dk + 3 or dk + 4and with no critical cell otherwise.We now define a matching on ψ − ( c d − ). Idea here is similar to that of step 2. Step d − : Observe that, in C n , there are exactly d − d − d , and these subgraphs are path graphs of length d − i.e., one of theelement of the following set: L = (cid:8) L { , ,...,d − ,d } d − , L { , ,...,d − ,d,d +1 } d − , . . . , L { d,d +1 ,..., d − , d − } d − (cid:9) .Thus, cells of ψ − ( c d − ) can be partitioned into d − L for each ∈ L . Here, ∆ L is collection of all those cells σ ∈ ψ − ( c d − ) such that L is the connectedcomponent of C n [ σ ]. Clearly, ψ − ( c d − ) = F L ∈ L ∆ L . Now, the idea is to define acyclic matchingson ∆ L for each L ∈ L and merge them together to get an acyclic matching on ψ − ( c d − ).(1) Observe that, for p ≥ d −
2, the p -cells of ∆ L { , ,...,d − ,d } d − are in 1-1 correspondence withthe p − ( d −
2) cells of Ind d − ( P { d +2 ,...,n, } n − d ) with one extra cell of dimension d − { , , . . . , d − , d } . Using Proposition 4.3, let M be a perfect matching onInd d − ( P { d +2 ,...,n, } n − d ). Define a matching M d − on ∆ L { , ,...,d − ,d } d − as follows: ( σ, τ ) ∈ M iff ( σ ∪{ , , . . . , d − , d } , τ ∪{ , , . . . , d − , d } ) ∈ M d − . Therefore, we get the following.Matching M d − is an acyclic matching on ∆ L { , ,...,d − ,d } d − with the following property.If n − d = dk − dk , i.e., n = d ( k + 1) − d ( k + 1), then there is only one criticalcell of dimension dk − k − d − d ( k + 1) − k + 1) − { , , . . . , d − , d } ⊔ k − G i =1 { id + 3 , . . . , ( i + 1) d } ⊔ { dk + 3 , . . . , n, } , if n = d ( k + 1) − , { , , . . . , d − , d } ⊔ k − G i =1 { id + 3 , . . . , ( i + 1) d } ⊔ { dk + 3 , . . . , n } , if n = d ( k + 1) . (7) Otherwise, there is no critical cell.(2) We now define a matching on ∆ L { t,t +1 ,...,d + t − } d − for each t ∈ { , , . . . , d } . Similar to thecase of ∆ L { , ,...,d − ,d } d − , we define an acyclic matching on ∆ L { t,t +1 ,...,d + t − } d − , say M td − usingthe perfect matching defined on Ind d − ( P { d +2 ,...,n, } n − d ). We thus get the following.If n − d = dk − dk , i.e., n = d ( k + 1) − d ( k + 1), then there is only onecritical cell of dimension dk − k − d − d ( k + 1) − k + 1) − { t, t + 1 , . . . , d + t − } ⊔ k − G i =1 { id + t, . . . , ( i + 1) d + t − } ⊔ { dk + t, . . . , n, , . . . , t − } , if n = d ( k + 1) − { t, t + 1 , . . . , d + t − } ⊔ k − G i =1 { id + t, . . . , ( i + 1) d + t − } ⊔ { dk + t, . . . , n, , . . . , t − } , if n = d ( k + 1) . (8) Otherwise, there is no critical cell.Since ψ − ( c d − ) = F L ∈ L ∆ L , M d − = d F t =3 M td − (defined in step d −
2) is an acyclic matchingon ψ − ( c d − ) with exactly d − d ( k + 1) − k + 1) − n = d ( k + 1) − d ( k + 1) and with no critical cell otherwise.Using Theorem 2.10, we observe that M = d − F i =1 M i is an acyclic matching on φ − ( e d ) with: • no critical cell if n = dk + 1, • exactly 1 critical cell of dimension dk − k if n = dk + 2 • exactly t − dk − k + t − t − dk − k + t −
2, if n = dk + t for some t ∈ { , . . . , d − }• exactly d − d ( k + 1) − k + 1) − n = d ( k + 1).We now define another matching on the set of critical cells corresponding to matching M on φ − ( e d ). The Idea is the following. If n = dk + 3, then observe from step 1 and step 2 that if γ is critical of dimension dk − k then γ ∪ { d − } is critical of dimension dk − k + 1. So match γ with γ ∪ { d − } . Now, let n = dk + t for some t ∈ { , . . . , d − } . From step t − − t − γ = { d − i, . . . , d, . . . , d + t − i − } ∪ { β } is a critical cell ofdimension dk − k + t − t − { d − i − , d − i, . . . , d, . . . , d + t − i − } ∪ { β } is critical cell of dimension dk − k + t −
2. Here, we match γ with γ ∪ { d − i − } . Let thematching defined above is M ′ . Claim 4.
Let M and M ′ be matchings on φ − ( e d ) as defined above. Then, M = M ⊔ M ′ is anacyclic matching on φ − ( e d ) with • no critical cell if n = dk + 1 , • exactly critical cell of dimension dk − k +1)+ t if n = dk + t for some t ∈ { , . . . , d − } , • exactly d − critical cells of dimension d ( k + 1) − k + 1) − if n = d ( k + 1) ,Proof of Claim 4. Let ∆ = { σ ∈ φ − ( e d ) : σ ∈ η for some η ∈ M } and ∆ = φ − ( e d ) \ ∆ .Since M and M ′ are union of a sequence of elementary matchings on ∆ and ∆ respectively, M and M ′ are acyclic matching from Proposition 2.9.Further, it is clear from the description of the critical cells given in step-1 to step-( d − τ ∈ ∆ and σ ∈ ∆ then τ * σ . Thus, using Theorem 2.11, we get that M is an acyclicmatching on φ − ( e d ). Calculation of number of critical cells corresponding to matching M isstraight forward once we fix an n . (cid:3) Case 3:
In cases 1 and 2, we defined acyclic matchings on φ − ( e id ) for i ∈ { , . . . , k } . Here,we consider the preimage φ − ( e r ) and define a matching M ′ on it. • If n = dk , then φ − ( e r ) is isomorphic to Ind d − ( G ), where G is isomorphic to the uinon k disjoint copies of path graphs of length d −
1. From Corollary 4.4, there exists an acyclicmatching on the face poset of Ind d − ( G ) with exactly one critical cell of dimension dk − k − • If n = dk + 1, then φ − ( e r ) is isomorphic to Ind d − ( G ), where G is isomorphic to theunion k − P d − and one copy of P d . Again from Corollary 4.4, thereexists an acyclic matching on the face poset of Ind d − ( G ) with exactly one critical cellof dimension dk − k − • If n = dk, dk + 1 then one connected component of C n \ { d, d, . . . , dk } will be a pathgraph of cardinality either less than d − d and less than 2 d −
2. Inboth the cases, using Corollary 4.5 there exists a matching on φ − ( c r ) with no criticalcell.From Eq. (3), Theorem 2.10, case (1), Claim 4 and case 3, we get that M ∪ M ′ is an acyclicmatching on F (Ind d − ( C n )) with • exactly d − dk − k −
1) if n = dk , • exactly one critical cell of dimension ( dk − k + t −
2) if n = dk + t for some t ∈{ , . . . , d − } .Hence, Theorem 4.6 follows from Corollary 2.5. (cid:3) The case of perfect m -ary trees For fixed m ≥
2, an m -ary tree is a rooted tree in which each node has no more than m children. A full m -ary tree is an m -ary tree where within each level every node has either 0 or m children. A perfect m -ary tree is a full m -ary tree in which all leaf nodes are at the samedepth (the depth of a node is the number of edges from the node to the tree’s root node).Following are some known facts about the perfect m -ary tree of height h , denoted B mh (seeFig. 5 for example).(1) B mh has h P i =0 m i = m h +1 − m − nodes.(2) For 0 ≤ t ≤ h , the number of nodes of depth t in B mh is m t .(3) B mh has m h leaf nodes.Before going into the computations of the homotopy type of r independence complexes of B mh , let us fix some notations. , a , a , a , a , a , a , (a) B a , a , a , a , a , a , a , a , a , a , a , a , a , (b) B Figure 5 • Let G be a graph and A ⊂ V ( G ). Then G [ A ] will denote the induced subgraph of G onvertex set A and G − A will denote the subgraph G [ V ( G ) \ A ]. • For d ∈ { , , . . . , h } , let V d ( B mh ) denote the set of vertices of B mh of depth d . • Let the vertices of B mh of depth d are represented by a d, , a d, , . . . , a d,m d from left toright (see Fig. 5). • The following ordering of the vertices of B mh will be used in the proofs of this section.Given a p,q , a p ′ ,q ′ ∈ V ( B mh ), we say that a p,q < a p ′ ,q ′ whenever q < q ′ and if q = q ′ then p < p ′ . For example, in B , a , < a , < a , < a , . • For σ ∈ ∆, denote σ ∪ { v } by σ ∪ v . Remark 5.1.
For simplicity of notations, B h will be denoted by B h . We first give some examples to explain our method for computing the homotopy type ofhigher independence complexes of B h . Example 5.2.
Here we compute the homotopy type of
Ind ( B ) . Define an element matchingon Ind ( B ) using the vertex a , as follows, M ( a , ) = { ( σ, σ ∪ a , ) : a , / ∈ σ, and σ, σ ∪ a , ∈ Ind ( B ) } , and N ( a , ) = { σ ∈ Ind ( B ) : σ ∈ η for some η ∈ M ( a , ) } . (9) Let ∆ = Ind ( B ) \ N ( a , ) . Observe that, if σ ∈ ∆( a , ) then σ ∪ a , / ∈ Ind ( B ) . By defini-tion of Ind r ( G ) , we observe that either { a , , a , , a , , a , } ⊆ σ or { a , , a , , a , , a , } ⊆ σ or { a , , a , , a , , a , } ⊆ σ . Since { a , , a , , a , , a , } , { a , , a , , a , , a , } , { a , , a , , a , , a , } are maximal cells of Ind ( B ) , these are the only unmatched cells i.e., ∆ = (cid:8) { a , , a , , a , , a , } , { a , , a , , a , , a , } , { a , , a , , a , , a , } (cid:9) . Therefore, Corollary 2.5 implies that Ind ( B ) ≃ W S . Example 5.3.
Using the homotopy type of
Ind ( B ) , we compute the homotopy type of Ind ( B ) .Here, we show that Ind ( B ) ≃ Ind ( B − { a , } ) . It is easy to see that B − { a , } ∼ = B ⊔ B .Thus, Observation 3.4 ( iv ) implies that Ind ( B ) ≃ Ind ( B ) ∗ Ind ( B ) ≃ W S .We now prove that Ind ( B ) ≃ Ind ( B − { a , } ) . Let R ( a , ) = { σ ∈ Ind ( B ) : a , ∈ σ } .Clearly, Ind ( B ) \ R ( a , ) = Ind ( B − { a , } ) . From Corollary 2.6, it is enough to define aperfect matching on R ( a , ) . We do so by defining a sequence of elementary matching usingvertices a , , a , , a , , a , as follows: Let ∆ = Ind ( B ) . For i ∈ { , , , } , define M ( a , i − ) = { ( σ, σ ∪ a , i − ) : a , ∈ σ, a , i − / ∈ σ and σ, σ ∪ a , i − ∈ ∆ i − ) } ,N ( a , i − ) = { σ ∈ ∆ i − : σ ∈ η for some η ∈ M ( a , i − ) } , and ∆ i = ∆ i − \ N ( a , i − ) . Claim 5. ∆ = Ind ( B ) \ R ( a , ) . ince N ( a , i − ) ⊆ R ( a , ) for all i ∈ { , , , } , Ind ( B ) \ R ( a , ) ⊆ ∆ . To show the otherway inclusion, it is enough to show that if σ ∈ Ind ( B ) and a , ∈ σ then σ ∈ N ( a , i − ) forsome i ∈ { , , , } .Let σ ∈ Ind ( B ) and a , ∈ σ . Since a , ∈ σ , it follows from the definition of Ind r ( G ) that { a , , a , , a , , a , } * σ . If { a , , a , } * σ , then σ ∈ N ( a , ) . If { a , , a , } ⊆ σ and a , / ∈ σ , then σ ∈ N ( a , ) . If { a , , a , , a , } ⊆ σ then a , / ∈ σ , implying that σ ∈ N ( a , ) .This completes the proof of Claim 5. To get the better understanding if the computations, we first prove our results for perfectbinary trees. The proof for perfect m -ary trees will follows using similar arguments. Lemma 5.4.
Let r ≥ h − . Then the homotopy type of r th independence complex of the graph B h is given as follows, Ind r ( B h ) ≃ W ( h − s ) S r − , if r = 2 h − s for some s ∈ { , , . . . , h − } , { point } , if r ≥ h +1 − . Proof.
The idea of the proof here is similar to that of in Example 5.2. If r ≥ h +1 −
1, thenObservation 3.4( i ) implies the result. Let r = 2 h − s for some fixed s ∈ { , , . . . , h − } and ∆ = Ind r ( B h ). Define a sequence of elementary matching using the alternate vertices ofdepth h , i.e. , a h, , a h, , . . . , a h, h − . For i ∈ { , , . . . , h − } , define M ( a h, i − ) = { ( σ, σ ∪ a h, i − ) : a h, i − / ∈ σ and σ, σ ∪ a h, i − ∈ ∆ i − } ,N ( a h, i − ) = { σ ∈ ∆ i − : σ ∈ η for some η ∈ M ( a h, i − ) } , and∆ i = ∆ i − \ N ( a h, i − ) . (10)We now show that the set of critical cells ∆ h − , corresponding to the sequence of matchingdefined in Eq. (10) is a set of (cid:0) h − s (cid:1) elements of fixed cardinality r . Thus, we get the resultusing Corollary 2.5. Claim 6. (1) If σ ∈ ∆ h − , then h − F j =0 V j ( B h ) ⊆ σ . (2) If σ ∈ ∆ h − , then σ is of cardinality r . (3) Cardinality of the set of critical cells ∆ h − is (cid:0) h − s (cid:1) .Proof of Claim 6. To the contrary of Claim 6(1), assume that there exists σ ∈ ∆ h − suchthat h − F j =0 V j ( B h ) * σ . Let a i ,j ∈ h − F j =0 V j ( B h ) be the smallest element with respect to the givenordering above such that a i ,j / ∈ σ . Since a i ,j is not a leaf, let a i ,j be the first children of a i ,j . Let a h,ℓ be the left most leaf of the sub-tree rooted at a i ,j . Further, the number ofvertices of sub-tree rooted at a i ,j is not more than 2 h −
1. Thus, σ ∈ N ( a h,ℓ ) (being the leftmost child of a sub-tree, ℓ is an odd number) contradicting the assumption that σ ∈ ∆ h − .This proves Claim 6(1).We now prove the second part of the above claim. Let σ ∈ ∆ h − . Clearly, cardinality of σ is at least r (because any cell of Ind r ( B h ) of cardinality less that r is in N ( a h, )). UsingClaim 6(1), we see that B h [ σ ] is connected graph of cardinality equal to the cardinality of σ .Therefore, the cardinality of σ is at most r . This proves Claim 6(2).From Eq. (10), it is clear that, if σ ∈ Ind r ( B h ) and a h, ∈ σ then σ ∈ N ( a h, ) implying that σ / ∈ ∆ h − . Hence, using Claim 6(1) and (2), we get that the cardinality of the set ∆ h − is equalto number of s -subsets of the set V h ( B h ) \ { a h, } . Which is equal to (cid:0) h − s (cid:1) . This completes theproof of Claim 6. (cid:3) rom Claim 6, we see that the matching on Ind r ( B h ) defined in Eq. (10) has (cid:0) h − s (cid:1) criticalcells of fixed dimension r −
1. Therefore, Lemma 5.4 follows from Corollary 2.5. (cid:3)
We are now ready to present the computation of homotopy type of Ind r ( B h ) for any r . Theorem 5.5.
For a fixed t ≥ , let r = 2 t − s for some s ∈ { , , . . . , t − } . Then the r th independence complex of the graph B h is given as follows, Ind r ( B h ) ≃ W p S q , if h = ( k − t + 2) + t + 1 for some k ≥ , W p S q , if h = k ( t + 2) + t for some k ≥ , { point } , otherwise , where, p = (cid:18) t − s (cid:19) +2 t +2 + ··· +2 ( k − t +2) ) and q = 2 r (2 + 2 t +2 + · · · + 2 ( k − t +2) ) − ,p = (cid:18) t − s (cid:19) +2 t +2 + ··· +2 k ( t +2) ,q = r (2 + 2 t +2 + · · · + 2 k ( t +2) ) − . Proof.
The idea here is similar to that of Example 5.3. If h ≤ t , then the result follows fromLemma 5.4. Let h > t . Here, we show that Ind r ( B h ) ≃ Ind r ( G ), where G is disjoint union ofperfect binary trees of height at most t . Recall that V j ( B h ) denotes the set of vertices of B h ofdepth j . Claim 7.
Ind r ( B h ) ≃ Ind r ( B h − V h − ( t +1) ( B h ))) .Proof of Claim 7. Let R ( V h − ( t +1) ( B h )) = { σ ∈ Ind r ( B h ) : σ ∩ V h − ( t +1) ( B h ) = ∅} . Clearly,Ind r ( B h ) \ R ( V h − ( t +1) ( B h )) = Ind r ( B h − V h − ( t +1) ( B h )). To prove Claim 7, from Corollary 2.6,it is enough to define a perfect matching on R ( V h − ( t +1) ( B h )). We do so by defining a sequenceof elementary matching on Ind r ( B h ) using vertices a h, , a h, , . . . , a h, h − as follows: Let ∆ =Ind r ( B h ). For i ∈ { , , . . . , h − } , define M ( a h, i − ) = { ( σ, σ ∪ a h, i − ) : σ ∩ V h − ( t +1) ( B h ) = ∅ , a h, i − / ∈ σ and σ, σ ∪ a h, i − ∈ ∆ i − } ,N ( a h, i − ) = { σ ∈ ∆ i − : σ ∈ η for some η ∈ M ( a h, i − ) } , and∆ i = ∆ i − \ N ( a h, i − ) . We now prove that ∆ h − = Ind r ( B h ) \ R ( V h − ( t +1) ( B h )). Which, along with Corollary 2.6,will imply Claim 7. Since N ( a h, i − ) ⊆ R ( V h − ( t +1) ( B h )) for all i ∈ { , , . . . , h − } , Ind r ( B h ) \ R ( V h − ( t +1) ( B h )) ⊆ ∆ h − . To show that ∆ h − ⊆ Ind r ( B h ) \ R ( V h − ( t +1) ( B h )), it is enoughto show that if σ ∈ Ind r ( B h ) and σ ∩ V h − ( t +1) ( B h ) = ∅ then σ ∈ N ( a h, i − ) for some i ∈{ , , . . . , h − } i.e. , σ / ∈ ∆ h − .Let σ ∈ Ind r ( B h ) such that σ ∩ V h − ( t +1) ( B h ) = ∅ . Without loss of generality, assume that a h − ( t +1) ,ℓ be the smallest vertex of V h − ( t +1) ( B h ) such that a h − ( t +1) ,ℓ ∈ σ . Let B ( a h − ( t +1) ,ℓ , B h )be the sub-tree of B h rooted at a h − ( t +1) ,ℓ . Let S denotes the set of all non-leaf vertices of B ( a h − ( t +1) ,ℓ , B h ), i.e. , S = t +1 F j =1 V h − j ( B h ) T V ( B ( a h − ( t +1) ,ℓ , B h )). Clearly, B ( a h − ( t +1) ,ℓ , B h ) is aperfect binary tree of height t + 1 and the cardinality of S is 2 t +1 −
1. Since B h [ S ] is a connectedgraph and r < t +1 − S * σ . Let a i ,j be the smallest element of S such that a i ,j / ∈ σ .Since a i ,j ∈ S and a h − ( t +1) ,ℓ ∈ σ , we get that i ∈ { h − t, h − t + 1 , . . . , h − } . Let a i +1 ,j be the left children of a i ,j and a h,ℓ be the left most leaf of perfect binary sub-tree rooted t a i +1 ,j . Observe that the cardinality of the sub-tree rooted at a i +1 ,j is at most 2 t − σ ∈ N ( a h,ℓ ) (here ℓ is an odd number because it is the left most leaf of a perfectbinary sub-tree of perfect binary tree). This completes the proof of Claim 7. (cid:3) We prove Theorem 5.5 using induction on h . Step : In this step, we prove the result for h ∈ { t + 1 , t + 2 , . . . , ( t + 2) + t } .From Claim 7, we see that Ind r ( B h ) ≃ Ind r ( B h − V h − ( t +1) ( B h ))). Observe that B h − V h − ( t +1) ( B h ) is disjoint union of 2(2 h − ( t +1) ) copies of perfect binary trees of height t and one perfect binary tree of height h − ( t + 2) (here, by B − we mean empty graph).Therefore, using Observation 3.4( iv ) and Lemma 5.4, we get the following equivalence.Ind r ( B h ) ≃ Ind r ( B t ⊔ · · · ⊔ B t | {z } h − ( t +1) )-copies ⊔ B h − ( t +2) ) ≃ Ind r ( B t ) ∗ · · · ∗ Ind r ( B t ) | {z } h − ( t +1) )-copies ∗ Ind r ( B h − ( t +2) ) ≃ Ind r ( B t ) ∗ Ind r ( B t ) ∗ Ind r ( B − ) , if h = t + 1 , Ind r ( B t ) ∗ · · · ∗ Ind r ( B t ) | {z } (2 t +2 )-copies ∗ Ind r ( B t ) , if h = ( t + 2) + t, Ind r ( B t ) ∗ · · · ∗ Ind r ( B t ) | {z } h − ( t +1) )-copies ∗{ point } , if t + 1 < h < ( t + 2) + t. (11) Thus, Lemma 5.4 and Lemma 2.2 implies the result, i.e. ,Ind r ( B h ) ≃ W ( t − s ) S r (2 ) − , if h = t + 1 , W ( t − s ) (20+2 t +2) S r (2 +2 t +2 ) − , if h = ( t + 2) + t, { point } , if t + 1 < h < ( t + 2) + t. Step : In this step, we prove the result for h ∈ { ( t + 2) + t + 1 , . . . , t + 2) + t } .Following similar method as in step 1, we get the following equivalence,Ind r ( B h ) ≃ Ind r ( B t ) ∗ · · · ∗ Ind r ( B t ) | {z } h − ( t +1) )-copies ∗ Ind r ( B h − ( t +2) )Observe that h − ( t + 2) is in { t + 1 , t + 2 , . . . , ( t + 2) + t } . Thus, result of Step 1implies the following. nd r ( B h ) ≃ Ind r ( B t ) ∗ · · · ∗ Ind r ( B t ) | {z } +2 t +2 )-copies ∗ Ind r ( B − ) , if h = ( t + 2) + t + 1 , Ind r ( B t ) ∗ · · · ∗ Ind r ( B t ) | {z } (2 t +2 +2 t +2) )-copies ∗ Ind r ( B t ) , if h = 2( t + 2) + t, Ind r ( B t ) ∗ · · · ∗ Ind r ( B t ) | {z } h − ( t +1) )-copies ∗{ point } , if ( t + 2) + t + 1 < h < t + 2) + t. ≃ Ind r ( B t ) ∗ · · · ∗ Ind r ( B t ) | {z } +2 t +2 )-copies , if h = ( t + 2) + t + 1 , Ind r ( B t ) ∗ · · · ∗ Ind r ( B t ) | {z } (2 +2 t +2 +2 t +2) )-copies , if h = 2( t + 2) + t, { point } , if ( t + 2) + t + 1 < h < t + 2) + t. Using Lemma 5.4 and Lemma 2.2, we get the result, i.e. ,Ind r ( B h ) ≃ W ( t − s ) t +2) S r (2 +2 t +2 ) − , if h = ( t + 2) + t + 1 , W ( t − s ) (20+2 t +2+22( t +2)) S r (2 +2 t +2 +2 t +2) ) − , if h = 2( t + 2) + t, { point } , if ( t + 2) + t + 1 < h < t + 2) + t. Step k : In this step, we prove the result for h ∈ { ( k − t + 2) + t + 1 , . . . , k ( t + 2) + t } where k ≥ r ( B h ) ≃ Ind r ( B t ) ∗ · · · ∗ Ind r ( B t ) | {z } h − ( t +1) )-copies ∗ Ind r ( B h − ( t +2) )Thus, result of Step k − r ( B h ) ≃ Ind r ( B t ) ∗ · · · ∗ Ind r ( B t ) | {z } +2 t +2 + ··· +2 ( k − t +2) )-copies ∗ Ind r ( B − ) , if h = ( k − t + 2) + t + 1 , Ind r ( B t ) ∗ · · · ∗ Ind r ( B t ) | {z } (2 t +2 +2 t +2)+ ··· +2 k ( t +2) )-copies ∗ Ind r ( B t ) , if h = k ( t + 2) + t, Ind r ( B t ) ∗ · · · ∗ Ind r ( B t ) | {z } h − ( t +1) )-copies ∗{ point } , if ( k − t + 2) + t + 1 < h < k ( t + 2) + t. ≃ Ind r ( B t ) ∗ · · · ∗ Ind r ( B t ) | {z } +2 t +2 + ··· +2 ( k − t +2) )-copies , if h = ( k − t + 2) + t + 1 , Ind r ( B t ) ∗ · · · ∗ Ind r ( B t ) | {z } (2 +2 t +2 +2 t +2)+ ··· +2 k ( t +2)) -copies , if h = k ( t + 2) + t, { point } , if ( k − t + 2) + t + 1 < h < k ( t + 2) + t. ence, using Lemma 5.4 and Lemma 2.2, we get the result, i.e. ,Ind r ( B h ) ≃ W ( t − s ) t +2+ ··· +2( k − t +2)) S r (2 +2 t +2 + ··· +2 ( k − t +2) ) − , if h = ( k − t + 2) + t + 1 , W ( t − s ) (20+2 t +2+22( t +2)+ ··· +2 k ( t +2)) S r (2 +2 t +2 +2 t +2) + ··· +2 k ( t +2) ) − , if h = k ( t + 2) + t, { point } , otherwise . This completes the proof of Theorem 5.5. (cid:3)
We are now ready to generalize Lemma 5.4 and Theorem 5.5 for perfect m -ary trees. Hence-forth, m ≥ Lemma 5.6.
Let r ≥ m h − m − . Then the homotopy type of r th independence complex of the graph B mh is given as follows, Ind r ( B mh ) ≃ W ( mh − s ) S r − , if r = m h − m − + s for some s ∈ { , , . . . , m h − } , { point } , if r ≥ m h +1 − m − . Proof.
The proof here is exactly similar to the proof of Lemma 5.4, but we explain some parthere as well for completeness. If r ≥ m h +1 − m − , then Observation 3.4( i ) implies the result. Let r = m h − m − + s for some fixed s ∈ { , , . . . , m h − } and ∆ = Ind r ( B mh ). Define a sequence ofelementary matching using the following vertices of depth h : a h, , a h,m +1 , . . . , a h,m ( m h − − . For i ∈ { , , . . . , m h − } , define M ( a h,mi − ( m − ) = { ( σ, σ ∪ a h,mi − ( m − ) : a h,mi − ( m − / ∈ σ and σ, σ ∪ a h,mi − ( m − ∈ ∆ i − } ,N ( a h,mi − ( m − ) = { σ ∈ ∆ i − : σ ∈ η for some η ∈ M ( a h,mi − ( m − ) } , and∆ i = ∆ i − \ N ( a h,mi − ( m − ) . (12)We now show that the set of critical cells ∆ m h − , corresponding to the sequence of matchingdefined in Eq. (12) is a set of (cid:0) m h − s (cid:1) cells of fixed dimension r − Claim 8. (1) If σ ∈ ∆ m h − , then h − F j =0 V j ( B mh ) ⊆ σ . (2) If σ ∈ ∆ m h − , then σ is of cardinality r . (3) Cardinality of the set of critical cells ∆ m h − is (cid:0) m h − s (cid:1) . Using exactly similar arguments as in the proof of Claim 6, we get the proof of Claim 8.From Claim 8, we see that the matching on Ind r ( B mh ) defined in Eq. (12) has (cid:0) m h − s (cid:1) criticalcells of fixed dimension r −
1. Therefore, Lemma 5.6 follows from Corollary 2.5. (cid:3)
We are now ready to present the main result of this section.
Theorem 5.7.
For a fixed t ≥ , let r = (cid:0) t − P i =0 m i (cid:1) + s = m t − m − + s for some s ∈ { , , . . . , m t − } .Then the r th independence complex of the graph B mh is given as follows, Ind r ( B mh ) ≃ W p S q , if h = ( k − t + 2) + t + 1 for some k ≥ , W p S q , if h = k ( t + 2) + t for some k ≥ , { point } , otherwise , here, p = (cid:18) m t − s (cid:19) m ( m + m t +2 + ··· + m ( k − t +2) ) and q = mr ( m + m t +2 + · · · + m ( k − t +2) ) − ,p = (cid:18) m t − s (cid:19) m + m t +2 + ··· + m k ( t +2) ,q = r ( m + m t +2 + · · · + m k ( t +2) ) − . Proof. If h ≤ t , then the result follows from Lemma 5.6. Let h > t . Here, we show thatInd r ( B mh ) ≃ Ind r ( G ), where G is disjoint union of perfect m -ary trees of height at most t .Recall that V j ( B mh ) denotes the set of vertices of B mh of depth j . Claim 9.
Ind r ( B mh ) ≃ Ind r ( B mh − V h − ( t +1) ( B mh ))) .Proof of Claim 9. Let R ( V h − ( t +1) ( B mh )) = { σ ∈ Ind r ( B mh ) : σ ∩ V h − ( t +1) ( B mh ) = ∅} . Clearly,Ind r ( B mh ) \ R ( V h − ( t +1) ( B mh )) = Ind r ( B mh − V h − ( t +1) ( B mh )). Thus, it is enough to define a perfectmatching on R ( V h − ( t +1) ( B mh )). We do so by defining a sequence of elementary matching onInd r ( B mh ) using vertices a h, , a h,m +1 , . . . , a h,m h − ( m − as follows: Let ∆ = Ind r ( B mh ). For i ∈ { , , . . . , m h − } , define M ( a h,mi − ( m − ) = { ( σ, σ ∪ a h,mi − ( m − ) : σ ∩ V h − ( t +1) ( B mh ) = ∅ , a h,mi − ( m − / ∈ σ and σ, σ ∪ a h,mi − ( m − ∈ ∆ i − } ,N ( a h,mi − ( m − ) = { σ ∈ ∆ i − : σ ∈ η for some η ∈ M ( a h,mi − ( m − ) } , ∆ i = ∆ i − \ N ( a h,mi − ( m − ) . Using simlar arguments as in the proof of Claim 7, we get that ∆ m h − = Ind r ( B mh ) \ R ( V h − ( t +1) ( B mh )). This completes the proof of Claim 9. (cid:3) We prove Theorem 5.7 using induction on h . Step : In this step, we prove the result for h ∈ { t + 1 , t + 2 , . . . , ( t + 2) + t } .From Claim 9, we see that Ind r ( B mh ) ≃ Ind r ( B mh − V h − ( t +1) ( B mh ))). Observe that B mh − V h − ( t +1) ( B mh ) is disjoint union of m ( m h − ( t +1) ) copies of perfect m -ary trees ofheight t and one perfect m -ary tree of height h − ( t + 2) (here, by B m − we mean emptygraph). Therefore, using Observation 3.4( iv ) and Lemma 5.6, we get the followingequivalence.Ind r ( B mh ) ≃ Ind r ( B mt ⊔ · · · ⊔ B mt | {z } m ( m h − ( t +1) )-copies ⊔ B mh − ( t +2) ) ≃ Ind r ( B mt ) ∗ · · · ∗ Ind r ( B mt ) | {z } m ( m h − ( t +1) )-copies ∗ Ind r ( B mh − ( t +2) ) ≃ Ind r ( B mt ) ∗ · · · ∗ Ind r ( B mt ) | {z } m -copies ∗ Ind r ( B m − ) , if h = t + 1 , Ind r ( B mt ) ∗ · · · ∗ Ind r ( B mt ) | {z } ( m t +2 )-copies ∗ Ind r ( B mt ) , if h = ( t + 2) + t, Ind r ( B mt ) ∗ · · · ∗ Ind r ( B mt ) | {z } m ( m h − ( t +1) )-copies ∗{ point } , if t + 1 < h < ( t + 2) + t. (13) hus, Lemma 5.6 and Lemma 2.2 implies the result, i.e. ,Ind r ( B mh ) ≃ W ( mt − s ) m S mr − , if h = t + 1 , W ( mt − s ) ( m mt +2) S r ( m + m t +2 ) − , if h = ( t + 2) + t, { point } , if t + 1 < h < ( t + 2) + t. Step : In this step, we prove the result for h ∈ { ( t + 2) + t + 1 , . . . , t + 2) + t } .Following similar method as in step 1, we get the following equivalence,Ind r ( B mh ) ≃ Ind r ( B mt ) ∗ · · · ∗ Ind r ( B mt ) | {z } m ( m h − ( t +1) )-copies ∗ Ind r ( B mh − ( t +2) )Observe that h − ( t + 2) is in { t + 1 , t + 2 , . . . , ( t + 2) + t } . Thus, result of Step 1implies the following.Ind r ( B mh ) ≃ Ind r ( B mt ) ∗ · · · ∗ Ind r ( B mt ) | {z } m ( m + m t +2 )-copies ∗ Ind r ( B m − ) , if h = ( t + 2) + t + 1 , Ind r ( B mt ) ∗ · · · ∗ Ind r ( B mt ) | {z } ( m t +2 + m t +2) )-copies ∗ Ind r ( B mt ) , if h = 2( t + 2) + t, Ind r ( B mt ) ∗ · · · ∗ Ind r ( B mt ) | {z } m ( m h − ( t +1) )-copies ∗{ point } , if ( t + 2) + t + 1 < h < t + 2) + t. ≃ Ind r ( B mt ) ∗ · · · ∗ Ind r ( B mt ) | {z } m ( m + m t +2 )-copies , if h = ( t + 2) + t + 1 , Ind r ( B mt ) ∗ · · · ∗ Ind r ( B mt ) | {z } ( m + m t +2 + m t +2) )-copies , if h = 2( t + 2) + t, { point } , if ( t + 2) + t + 1 < h < t + 2) + t. Using Lemma 5.6 and Lemma 2.2, we get the result, i.e. ,Ind r ( B mh ) ≃ W ( mt − s ) m ( m mt +2) S mr ( m + m t +2 ) − , if h = ( t + 2) + t + 1 , W ( mt − s ) ( m mt +2+ m t +2)) S r ( m + m t +2 + m t +2) ) − , if h = 2( t + 2) + t, { point } , if ( t + 2) + t + 1 < h < t + 2) + t. Step k : In this step, we prove the result for h ∈ { ( k − t + 2) + t + 1 , . . . , k ( t + 2) + t } where k ≥ r ( B mh ) ≃ Ind r ( B mt ) ∗ · · · ∗ Ind r ( B mt ) | {z } m ( m h − ( t +1) )-copies ∗ Ind r ( B mh − ( t +2) ) hus, result of Step k − r ( B mh ) ≃ Ind r ( B mt ) ∗ · · · ∗ Ind r ( B mt ) | {z } m ( m + m t +2 + ··· + m ( k − t +2) )-copies ∗ Ind r ( B m − ) , if h = ( k − t + 2) + t + 1 , Ind r ( B mt ) ∗ · · · ∗ Ind r ( B mt ) | {z } ( m t +2 + m t +2)+ ··· + mk ( t +2) )-copies ∗ Ind r ( B mt ) , if h = k ( t + 2) + t, Ind r ( B mt ) ∗ · · · ∗ Ind r ( B mt ) | {z } m ( m h − ( t +1) )-copies ∗{ point } , if ( k − t + 2) + t + 1 < h < k ( t + 2) + t. ≃ Ind r ( B mt ) ∗ · · · ∗ Ind r ( B mt ) | {z } m ( m + m t +2 + ··· + m ( k − t +2) )-copies , if h = ( k − t + 2) + t + 1 , Ind r ( B mt ) ∗ · · · ∗ Ind r ( B mt ) | {z } ( m + m t +2 + m t +2)+ ··· + mk ( t +2)) -copies , if h = k ( t + 2) + t, { point } , if ( k − t + 2) + t + 1 < h < k ( t + 2) + t. Hence, using Lemma 5.6 and Lemma 2.2, we get the result (recall that t is fixed), i.e. ,Ind r ( B mh ) ≃ W ( mt − s ) m ( m mt +2+ ··· + m ( k − t +2)) S mr ( m + m t +2 + ··· + m ( k − t +2) ) − , if h = ( k − t + 2) + t + 1 , W ( mt − s ) ( m mt +2+ m t +2)+ ··· + mk ( t +2)) S r ( m + m t +2 + m t +2) + ··· + m k ( t +2) ) − , if h = k ( t + 2) + t, { point } , otherwise . This completes the proof of Theorem 5.7. (cid:3) Concluding remarks
In this section, we list a few interesting questions and conjectures.6.1.
Universality of higher independence complexes.
It was shown in [8] that every sim-plicial complex arising as the barycentric subdivision of a CW complex may be represented asthe 1-independence complex of a graph. One can invistigate whether a similar statement holdsfor for all r -independence complexes. From the definition it is clear that Ind r ( G ) contains allsubsets of V ( G ) of cardinality at most r + 1 implying that Ind r ( G ) is always ( r − r -independence complexes of graphs are may have torsion. Let M s ( G ) denotes the s th generalised mycielskian of a graph G . Then,˜ H i (Ind ( M ( C ))) = Z if i = 3 , Z if i = 5 , . One can now ask the following question.
Question 1.
Given r ≥ and an ( r − -connected simplicial complex X , does there exists agraph G such that Ind r ( G ) is homeomorphic to X ? Trees.
Kawamura [13] computed the exact homotopy of 1-independence complexes of treesand showed that they are either contractible or homotopy equivalent to a sphere. In Section 5,it was shown that the homotopy type of higher independence complexes of m -ary trees is alsothat wedge of spheres. So, one might hope for a similar result for the class of all trees as well.In another project [7] with Samir Shukla, authors have determined the homotopy type ofInd r ( G ) for chordal graphs G (note that class of tress is a subclass of chordal graphs). A chordal graph is a graph in which every cycle on more than 3 vertices has a chord. Homotopy ype of 1-independence complexes of chordal graphs was studied by Kawamura in [14]. Here,we only announce our result, without proving it. Theorem 6.1 ([7]) . The higher independence complexes of chordal graphs are either contractibleor homotopy equivalent to a wedge of spheres.
However, the following question is still unanswered.
Question 2.
Given r ≥ and a trees T , find a formula for the number of spheres in thehomotopy decomposition of Ind r ( T ) ? Shellable higher independence complexes.
In [22], Woodroofe showed that 1-independencecomplexes of chordal graphs are vertex-decomposable (hence shellable [21, Theorem 1.2]).In a joint work [6] with Manikandan, we have indentified a few classes of graphs whose r -independence complexes are shellable. Here, we pose a few problems in this direction. Question 3.
For which classes of graphs, the higher independence complexes are shellable?
One might expect a positive answer to the following question.
Question 4.
Whether
Ind r ( G ) is vertex-decomposable for each r ≥ and chordal graph G ? There is also the case of chordal graphs.
Conjecture 6.2. If G is a chordal graph then Ind r ( G ) is shellable for all r . Grid graphs.
For m, n ≥
2, a rectangular grid graph , denoted G m,n is a graph with V ( G m,n ) = { ( i, j ) : i ∈ [ m ] , j ∈ [ n ] } as its vertex set and ( i, j ) is adjacent to ( i , j ) in G m,n if and only if either ‘ i = i and j = j + 1’ or ‘ j = j and i = i + 1’. In the last decade,1-independence complexes of grid graphs have studied in details (see [4, 5, 12] for more details).We have analysed the complex Ind r ( G ,n ) (for small values of n ) and also computed homologytheir of using SageMATH [20] (see Table 1 below). Based on our calculations, we make thefollowing conjecture. Conjecture 6.3.
For all r ≥ n , Ind r ( G ,n ) is either contractible or homotopy equivalent to awedge of spheres of dimension r − . From Table 1, we also see that ˜ H i ( G , ) is non-trivial in two different dimensions (the notation i : Z p means ˜ H i (Ind r ( G ,n )) = Z p ). This raises the following question. Question 5.
What is the homotopy type of higher independence complexes of grid graphs G m,n ? n r Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z
11 : Z Z Z Z Z ; 7 : Z Z Z Z
11 : Z
13 : Z Z Table 1.
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Chennai Mathematical Institute, India
E-mail address : [email protected] Chennai Mathematical Institute, India
E-mail address : [email protected]@cmi.ac.in