Discrete Morse theory on graphs

Topology and its Applications 156 (2009) 3091–3100 Contents lists available at ScienceDirect Topology and its Applications www.elsevier.com/locate/topol Discrete Morse theory on graphs ✩ R. Ayala, L.M. Fernández, D. Fernández-Ternero, J.A. Vilches ∗ Dpto. de Geometría y Topología, Universidad de Sevilla, 41080, Sevilla, Spain a r t i c l e i n f o a b s t r a c t Article history: Received 15 October 2008 Received in revised form 22 December 2008 Accepted 26 January 2009 MSC: 57M20 57Q15 68R10 We characterize the topology of a graph in terms of the critical elements of a discrete Morse function defined on it. Besides, we study the structure and some properties of the gradient vector field induced by a discrete Morse function defined on a graph. Finally, we get results on the number of non-homologically equivalent excellent discrete Morse functions defined on some kind of graphs.  2009 Elsevier B.V. All rights reserved. Keywords: Infinite locally finite graph Critical element Gradient vector field Gradient path 1. Introduction Discrete Morse theory was introduced by R. Forman [5] as a purely combinatorial version of classical or smooth Morse theory. This approach has proven to be a powerful tool to study the topology of a general c w-complex. In our point of view, discrete Morse theory has two basic advantages over the smooth setting: mainly due to its discrete nature it obtains analogous results to the classical one but in a straightforward and less complicated way [5] and besides, it turns out to be very suitable to adapt results in a computational way [7]. There is a growing number of researchers who are finding different applications of this theory to solve problems in many areas, from denoising digital data sets [6,9], to establishing links with complexes of graphs [3] just to cite some of them. The authors are particularly interested in the extension of this theory to the non-compact case. In this sense we have obtained a generalized version of Morse inequalities for infinite graphs [1] and for triangulated and non-compact surfaces [2]. In this work the decreasing monotonous behaviour of a discrete Morse function at the ends plays an outstanding role, in fact, they are acting as a kind of critical simplices at the end of the considered complex and this is the reason to unify both concepts with the notion of critical element. The goal of this paper is to study some aspects of the extension of discrete Morse theory to the non-compact 1-dimensional case, namely, the extension to infinite graphs. We begin presenting in Section 2 the basic notions and results concerning infinite discrete Morse theory on graphs. In Section 3 we introduce the notions of excellent discrete Morse function, homology equivalence for this kind of functions and homological sequences. Section 4 is devoted to the study of ✩ * The authors are partially supported by the Plan Nacional de Investigación 2.007, Project MTM2007-65726, Spain, 2008. Corresponding author. E-mail addresses: [email protected] (R. Ayala), [email protected] (L.M. Fernández), [email protected] (D. Fernández-Ternero), [email protected] (J.A. Vilches). 0166-8641/$ – see front matter doi:10.1016/j.topol.2009.01.022  2009 Elsevier B.V. All rights reserved. 3092 R. Ayala et al. / Topology and its Applications 156 (2009) 3091–3100 the structure of the set of all gradient paths induced by a discrete Morse function defined on a graph. This is the discrete analogue of the smooth notion of flow lines of the field induced by a Morse function. By using the notion of tree rooted in a 0-critical element, we decompose this set as a disjoint union of such trees and hence, it can be seen as a forest. Later on, we include in Section 5 the exposition and proof of a result which characterizes those (infinite) graphs which admit a discrete Morse function with a given number of critical elements. Our infinite version of the discrete Morse inequalities and an explicit definition of the desired function are the basic tools that we use to prove it. Notice that this result is looking for the basic goal of classical Morse theory, that is, getting links between the topology of a manifold and the critical points of a smooth Morse function defined on it. Besides, in this section we introduce the notion of optimal discrete Morse function defined on an infinite graph and we prove that every graph admits an optimal discrete Morse function. Finally, Section 6 is devoted to carry out in the discrete setting a study which has been done in the smooth setting by Nicolaescu [11] for S 1 . Our goal in this section is counting the number of homology classes of excellent discrete Morse functions defined on a graph with a given number of critical simplices in the cases that the graph is a tree, a finite wedge of cycles or the union of a finite wedge of cycles and a tree. 2. Preliminaries Through all this paper, we only consider infinite graphs which are locally finite. Given such a graph G, we introduce here the basic notions of discrete Morse theory [5]. A discrete Morse function is a function f : G → R such that, for any p-simplex σ ∈ G: (M1) card{τ ( p +1) > σ / f (τ )  f (σ )}  1. (M2) card{υ ( p −1) < σ / f (υ )  f (σ )}  1. A p-simplex σ ∈ G is said to be a critical simplex with respect to f if: (C1) card{τ ( p +1) > σ / f (τ )  f (σ )} = 0. (C2) card{υ ( p −1) < σ / f (υ )  f (σ )} = 0. A value of a discrete Morse function on a critical simplex is called critical value. Given c ∈ R the level subcomplex G (c ) is the subcomplex of G consisting of all simplices of their faces, that is, G (c ) =  f (τ )c  τ with f (τ )  c, as well as all σ. σ τ A ray is an infinite sequence of simplices v 0 , e 0 , v 1 , e 1 , . . . , v r , e r , v r +1 , . . . verifying that the 0-simplices v i and v i +1 are faces of the 1-simplex e i , for any i ∈ N ∪ {0}. Two rays contained in an infinite graph are said to be equivalent or cofinal if both coincide from a common 0-simplex. If there is a discrete Morse function f defined on G, a decreasing ray is a ray satisfying f ( v 0 )  f (e 0 ) > f ( v 1 )  f (e 1 ) > · · ·  f (er ) > f ( v r +1 )  · · · . A critical element of f on G is either a critical simplex or a decreasing ray. Given a discrete Morse function defined on G, we say that a pair of simplices ( v < e ) is in the gradient vector field induced by f if and only if f ( v )  f (e ). Given a gradient vector field V on G, a V -path is a sequence of simplices α0( p) , β0( p+1) , α1( p) , β1( p+1) , . . . , βr( p+1) , αr(+p)1 , . . . , ( p) such that, for each i  0, the pair (αi ( p +1) < βi (1) ( p +1) ) ∈ V and βi ( p) ( p) > αi +1 = αi . Two discrete Morse functions f and g defined on a simplicial complex M are equivalent if every pair of simplices and β ( p +1) in M such that α ( p ) < β ( p +1) verify that f (α ) < f (β) if and only if α ( p) g (α ) < g (β). The next result states that any two equivalent discrete Morse functions have the same gradient vector field and conversely. Theorem 2.1. ([4]) Two discrete Morse functions f and g defined on a simplicial complex M are equivalent if and only if f and g induce the same gradient vector field. R. Ayala et al. / Topology and its Applications 156 (2009) 3091–3100 3093 Given a discrete vector field V defined on an (infinite) graph, it is easy to prove that if V does not contains any closed V -path then, there exits a (proper) discrete Morse function (in fact, infinitely many) such that V = V f . Following the main goal of classical Morse theory, that is, looking for links between the topology of a manifold and the critical points of a Morse function defined on it, the authors proved the following result, which generalizes to the infinite 1-dimensional case the well known Morse inequalities. Theorem 2.2. ([1]) Let G be an infinite graph and let f be a discrete Morse function defined on G such that the numbers of critical i-simplices of f with i = 0, 1, denoted by mi ( f ), are finite and the number of non-cofinal decreasing rays, denoted by d0 , is finite too. Then: (i) m0 ( f ) + d0  b0 and m1 ( f )  b1 , where b i denotes the ith Betti number of G with i = 0, 1. (ii) b0 − b1 = m0 ( f ) + d0 − m1 ( f ). 3. Excellent discrete Morse functions on graphs As we will see in Sections 4 and 5, the topological properties of a graph are deeply related to the qualitative properties of the discrete Morse functions defined on it. Roughly speaking, these properties are essentially the number of critical elements of the functions and the changes on the topology of their level subcomplexes which are detected by the behaviour of its Betti numbers. Thus, it seems reasonable to consider two discrete Morse functions defined on a graph as indistinguishable if these data sets are the same for both functions. For this reason, it is convenient to deal with functions whose critical values, that is, its values on the critical simplices, are different and we assume that two such functions are equivalent if their level subcomplexes have the same homology. Definition 3.1. A discrete Morse function is called excellent if all its critical values are different. Definition 3.2. Two excellent discrete Morse functions f and g defined on G with critical values a0 < a1 < · · · < am−1 and c 0 < c 1 < · · · < cm−1 respectively will be called homologically equivalent if for all i = 0, . . . , m − 1 the level subcomplexes G (ai ) and G (c i ) have the same Betti numbers. Definition 3.3. Let f be an excellent discrete Morse function defined on a connected graph G with m critical simplices and critical values a0 , . . . , am−1 . We denote the level subcomplexes G (ai ) by G i for all i = 0, 1, . . . , m − 1. The homological sequences of f are the two sequences B 0 , B 1 : {0, 1, . . . , m − 1} → N containing the homological information of the level subcomplexes G 0 , . . . , G m−1 , that is, B 0 (i ) = b0 (G i ) = dim( H 0 (G i )) and B 1 (i ) = b1 (G i ) = dim( H 1 (G i )) for each i = 0, 1, . . . , m − 1. Remark 3.4. The homological sequences of f satisfy B 0 (0) = B 0 (m − 1) = b0 = 1, B 1 (0) = 0, B 1 (m − 1) = b1 , B 0 ( i ) > 0,    B 0 (i + 1) − B 0 (i ) = 0 or 1; B 1 (i + 1) − B 1 (i ) = 0 or 1. Lemma 3.5. For each i = 0, 1, . . . , m − 2 it holds one and only one of the following identities: (H1) B 0 (i ) = B 0 (i + 1). (H2) B 1 (i ) = B 1 (i + 1). Proof. Since every interval (ai , ai +1 ] contains a unique critical value, the level subcomplexes G i and G i +1 are homologically different, but only one of the Betti numbers of G i and G i +1 are the same. ✷ It is interesting to point out that identity (H2) of the above lemma reveals the appearance of a new connected component or the join of two different connected components in the process of obtention of G by level subcomplexes. Analogously, identity (H1) reveals the creation of a new 1-cycle of G in this process. Notice that two excellent discrete Morse functions are homologically equivalent if and only if their homological sequences are the same. 4. The gradient vector field of a discrete Morse function on a graph The qualitative properties of a discrete Morse function f are reflected by its induced gradient vector field V f . Thus, in many situations we do not need to consider the values of f but we just deal with that field. In fact, the authors proved in [4] that two different discrete Morse functions f and g defined on a graph verify V f = V g if and only if they have the same sets of critical elements. Note that this result is no longer true for locally finite simplicial complexes with dimension 3094 R. Ayala et al. / Topology and its Applications 156 (2009) 3091–3100 greater than or equal to two. Moreover, the above characterization is equivalent to the following condition: If σ < τ then f (σ ) < f (τ ) if and only if g (σ ) < g (τ ). Now we are going to study the structure of the gradient field of a discrete Morse function f defined on a graph G. It can be easily proved that a gradient vector field V does not contain closed V -paths. Definition 4.1. Given a 0-critical element in G, that is, a critical vertex v or a decreasing ray r, we say that a vertex w of G is rooted in v (respectively in r) if there exists a finite V -path joining w and v (respectively w and some vertex of r). Note that if a vertex w is rooted in a decreasing ray r, then there exists a decreasing ray r ′ starting from w which is equivalent to r. Proposition 4.2. Let G be an infinite graph and let f be a discrete Morse function defined on G. It holds that: 1. Given w any vertex of G, there is a unique 0-critical element on which w is rooted. 2. Given any 0-critical element (v or r), the set of all V -paths rooted in it is a tree called the tree rooted in v or r and denoted by T v or T r . 3. Any two of such rooted trees are disjoint. Proof. 1. Let us suppose that there exists a vertex w of G rooted in two 0-critical elements p and q (these critical elements can be vertices or decreasing rays). Then there exist two finite V -paths we 1 v 1 e 2 v 2 · · · er v r and w ē 1 u 1 ē 2 u 2 · · · ē s u s where v r = p or a vertex in p if p is a decreasing ray and u s = q or a vertex in q if q is a decreasing ray. As f is a discrete Morse function, it is not possible that both f ( w )  f (e 1 ) and f ( w )  f (ē 1 ) if e 1 = ē 1 . Therefore e 1 = ē 1 and v 1 = u 1 , and reasoning in the same way in each vertex we obtain that the two V -paths are the same. 2. Let p be a 0-critical element and let us consider the union T p of all V -paths rooted in p. If there exists a cycle C in T p for some vertex v in C there would exist two different V -paths joining v with p, but this is not possible by (1). Thus, T p is a tree since it contains no cycle. 3. If there exist two trees rooted in different 0-critical elements p and q such that T p ∩ T q = ∅, any vertex in the intersection would be rooted in both p and q and that is not possible by (1). ✷ Remark 4.3. If F is the union of all rooted trees in G, it is easy to prove that F is a tree if and only if there is a unique 0-critical element of f in G. Theorem 4.4. Under the above definitions and notations, F can be obtained by removing all critical edges of f on G. Proof. Let e 1 , . . . , em1 be the 1-critical elements of the discrete Morse function f on an infinite graph G and we set H = G − {e 1 , . . . , em1 }. As F and H are spanning subgraphs of G, to prove that F = H is enough to prove that these subgraphs have the same edges. If e is an edge in F , it is not possible that e = e i for any i = 1, . . . , m1 since e i cannot be in a V -path because f (e i ) is greater than the values of f on the vertices of e i . Then e is an edge of H . Conversely, if e is an edge in H , e is not a critical edge (e = e i for every i = 1, . . . , m1 ). Then, if e = uv it is not possible both f (u )  f (e ) and f ( v )  f (e ). We can suppose, for example, that f (u )  f (e ) > f ( v ). So we have that u and v are rooted in the same 0-critical element p because adding u and e to the V -path joining v with p we get a V -path joining u with p. Thus, e is an edge of F since e is in the tree rooted in p and F is the union of all rooted trees. ✷ Definition 4.5. Let f be a discrete Morse function defined on an infinite graph G with m critical elements. The forest generated by r 0-critical elements v 1 , . . . , v r and s superfluous critical edges e 1 , . . . , e s is the forest consisting of: • the trees T v i with 1  i  r, and • the edges e i with 1  i  s. Remark 4.6. Observe that the forest F is the forest generated by all the 0-critical elements of f . 5. The critical elements of a discrete Morse function on a graph Once the generalized Morse inequalities have been introduced, we are in condition to extend the notion of optimality for discrete Morse function defined on infinite graphs. Classically optimal Morse functions are those on which Morse inequalities became equalities. We shall use this idea in the next definition. Definition 5.1. Let f be a discrete Morse function defined on an (infinite) graph G. We say that f is optimal if m0 ( f ) + d0 = b0 (G ) and m1 ( f ) = b1 (G ). R. Ayala et al. / Topology and its Applications 156 (2009) 3091–3100 3095 In fact, as an easy consequence of Theorem 2.2 we have that the two conditions of the above definition can be reduced to just any of them as the next result states. Proposition 5.2. Let f be a discrete Morse function defined on an (infinite) graph G. The following conditions are equivalent: (i) f is optimal. (ii) m0 ( f ) + d0 = b0 (G ). (iii) m1 ( f ) = b1 (G ). Proposition 5.3. Every connected graph G admits an optimal discrete Morse function. Proof. We will define a discrete Morse function on G with m = b1 (G ) + 1 critical elements. It is known (see [10]) that if T is a spanning tree in G there exists a bijection between the set of basic cycles of G and the set of b1 (G ) edges e 1 , . . . , eb1 not in T . First we take a spanning tree T of G. If we choose a root vertex v 0 in T , let f : T → R be defined as follows • if v is a vertex in the level t of T , we set f ( v ) = t, and • if uv is an edge in T , we set f (uv ) = max{ f (u ), f ( v )}. Now, if we extend f to each edge e i = u i v i , 1  i  b1 , by defining f (e i ) such that f (e i ) > max{ f (u i ), f ( v i )}, we obtain a discrete Morse function f on G whose critical elements are v 0 and the edges e 1 , . . . , eb1 . ✷ Taking into account the values of a discrete Morse function defined on a graph, it is interesting to consider two different kinds of critical simplices. Basically, we want to distinguish between those critical simplices which arise forced by the topology of the considered graph and those which are introduced by the non-optimality of the function. Given a nonoptimal discrete Morse function, by considering the ordered family of level subcomplexes associated to its critical values, we can control how a critical edge arises, that is, either it appears because it is completing a homology cycle or it appears due to the need to join two different connected components. Definition 5.4. Let f be an excellent discrete Morse function defined on a connected graph G with critical values a0 < · · · < an−1 . We say that a critical vertex v is an essential vertex if f ( v ) is the global minimum of f on G, that is, f ( v ) = a0 . One critical edge e i with f (e i ) = ai is an essential edge if B 1 (i ) − B 1 (i − 1) = 1. Otherwise, if a critical simplex is not an essential one, we say that it is a superfluous or cancellable simplex. Notice that it is straightforward to prove that a critical edge e i is essential if and only if e i is completing a 1-cycle which represents a basic element of H 1 (G ) not considered until this point. It interesting to point out that the concepts of cancellable and essential critical simplices are strongly matched to the topology of the studied graph G. So in this sense, we can say that essential critical simplices are those whose existence is forced by the homology groups of G. These ideas are going to be made precise in the following results. Proposition 5.5. Let f be a discrete Morse function defined on a connected graph G such that b1 < +∞. If e is an essential critical edge of f on G with vertices v and w, then there exists a spanning tree T in G such that e ∈ / T and e +  v w is a basic cycle of H 1 (G ), v w is the unique path joining v and w in T . where  Proof. By means of Theorem 4.4, it holds that if all critical edges of f on G are removed, then we obtain a forest. Since every essential critical edge is given by every independent cycle of H 1 (G ), it follows that if we just remove all essential critical edges, then we do not obtain any new connected component and hence, we get a spanning tree. Since there is a bijection between the independent cycles of H 1 (G ) and those edges of G such that do not belong to a spanning tree contained in G (see [10]), we obtain that every essential edge characterizes an independent cycle of H 1 (G ). This is precisely the 1-cycle obtained by gluing the essential critical edge e with the two gradient paths starting from e and merging at some vertex. ✷ Proposition 5.6. Let G be a connected graph with b1 < +∞. Then a discrete Morse function f on G is optimal if and only if all its critical edges are essential. Proof. By Proposition 5.5 the number of essential edges of f is b1 and by Proposition 5.2 the optimality condition is equivalent to m1 = b1 . Thus, we conclude that f is optimal if and only if there are no critical edges except the essential ones. ✷ 3096 R. Ayala et al. / Topology and its Applications 156 (2009) 3091–3100 The next result shows us the topological consequences of the existence of a discrete Morse function with no critical simplices. Proposition 5.7. Let G be a connected graph. If G admits a discrete Morse function with no critical simplices then G is an infinite tree. Proof. Let f be a discrete Morse function on G with no critical simplices. Then by Theorem 2.2 we have 0 + d0 = m0 ( f ) + d0  b0 = 1 and 0 = m1 ( f )  b1 . Thus, b1 = 0 and so G is a tree. Moreover, from b0 − b1 = m0 ( f ) + d0 − m1 ( f ), we also have 1 = d0 , that is, f has one decreasing ray on G, so that G must be infinite. ✷ Remark 5.8. If we only assume m1 = 0, we get that G is a forest. Now, we are going to characterize a graph by taking into account the total number of critical elements of a discrete Morse function defined on it. Theorem 5.9. An infinite connected graph G admits a discrete Morse function with m critical elements if and only if : (i) If m is odd, then G is either a tree or G = C ∪ F , where C is a finite subgraph containing 2h independent cycles which are a basis for H 1 (G ) with h  ⌈ m ⌉, F is a finite forest with at least an infinite tree and every tree in F intersects C in a unique vertex. 2 (ii) If m is even, G = C ∪ F where C is a finite subgraph containing 2h + 1 independent cycles which are a basis for H 1 (G ) with ⌉ − 1 and F is a forest in the same conditions as above. h  ⌈m 2 Proof. Let us assume that m is odd (the case m is even is analogous). So, we may assume that m = 2 j + 1 = m0 + d0 + m1 . Now, by means of Theorem 2.2, we get that 1 − b1 = m0 + d0 − m1 and adding both equalities we get that b1 = 2 j + 2 − 2(m0 + d0 ) = 2h. Hence, b1 is even. Moreover, by using Theorem 2.2, it holds that m0 + d0  1 which implies that ⌉. 0  b1  2 j, that is, 0  h  j = ⌈ m 2 Conversely, let us suppose that G is the union of 2h independent cycles C and a forest F . By means of the proof of Proposition 5.3, we can obtain an optimal discrete Morse function f on G with m = 1 + 2h. If m = 1 + 2 j > 1 + 2h, then it is possible to get a new (non-optimal) discrete Morse function  f starting from f and introducing j − h new pairs of critical vertices and edges. It can be done by selecting a non-critical edge e of f on G. Thus, one of its two bounding vertices v f = f on G − { v , e } and  f (e ) >  f ( v ) so  f has two new critical simplices: v and e. By satisfies f (e )  f ( v ). Then we define  repeating this procedure j − h times, we finally get the desired function. ✷ Remark 5.10. Notice that m = 1 implies that G is a tree. 6. Counting the number of discrete Morse functions on a graph In this section we shall obtain the number of elements of the set of classes of homologically equivalent discrete Morse functions on certain graphs. In the differentiable setting, this calculation was done for S 1 in [11]. Theorem 6.1. The number of homology equivalence classes of excellent discrete Morse functions with m = b0 + b1 + 2k critical elements on a graph G is: 1. C k if G is a tree, m−1 b1 1 2. C k 2k if G = S ∪ T 1 ∪ · · · ∪ T r where the T i are trees, 3. C k m−2 2k if G = b1 2k S1; 1 denotes the kth Catalan number, where C k = k+ 1 k b1 1 tree T i intersects S in a unique vertex. b1 S 1 denotes the union of b1 copies of S 1 by a common vertex and every Proof. 1. If G is a tree, then b0 = 1 and b1 = 0. Therefore B 1 is a sequence of zeros. By Lemma 3.5 we have B 0 (i + 1) = B 0 (i ) for every i, then B 0 (i + 1) − B 0 (i ) = ±1 for each i = 0, 1, . . . , m − 2. So, the sequence B 0 is as a walk in Z>0 starting and ending at 1, with length m − 1 = 2k and steps of size ±1. But it is known (see [8]) that the number of such walks is the   1 2k . So there are at most C k homology equivalence classes in this case. kth Catalan number C k = k+ 1 k In order to prove that there are exactly C k classes, we will construct an excellent discrete Morse function f on G such that its sequence B 0 is equal to a walk n0 , n1 , . . . , n2k . First, by subdividing sufficiently many times we can get that the number of simplices is greater than m. Next, we choose the m simplices which will be the critical elements of the Morse function: we select k + 1 0-simplices v 1 , . . . , v k+1 and we take k 1-simplices el in the following way: If the unique path joining two 0-simplices v i and v j does not contain any other R. Ayala et al. / Topology and its Applications 156 (2009) 3091–3100 3097 selected 0-simplex, el is some 1-simplex in this path. We shall denote the sets of selected vertices and edges by V c and E c respectively. Notice that if we remove the k selected 1-simplices of G, we obtain a forest F with k + 1 trees and each of these trees contains exactly one selected 0-simplex. So, for each 0-simplex v i , we can consider the tree of F containing v i as a tree rooted in v i , denoted T v i . Observe that, by means of by Proposition 4.2, once we have constructed the Morse function on G, the tree T v i is equal to the tree rooted in the 0-simplex v i . We will get the function f following the next steps: We begin with one 0-simplex p 0 ∈ V c which will be the global minimum of f and we define f ( p 0 ) = 0. Next, we define f on the tree T p 0 by levels as we did in the proof of Proposition 5.3 obtaining a Morse function on the tree T 0 = T p 0 , with one critical element p 0 and whose sequence B 0 has only one element which is n0 = 1. Step 1 As we have n1 = 2, we take a new 0-simplex p 1 ∈ V c such that there is a unique 1-simplex in E c contained in the unique path joining p 0 and p 1 . We set f ( p 1 ) = f ( p 0 ) + 1 and next, we define f on T p 1 by levels as it was done in Step 0. Thus, we get an excellent discrete Morse function on the subgraph T 1 = T p 0 ∪ T p 1 of G whose associated sequence is B 0 = (n0 , n1 ) = (1, 2), since T 1 is not connected because T p 0 ∩ T p 1 = ∅. Step j + 1 Suppose that we have already defined f on the subgraph Step 0 T j = T p 0 ∪ T p 1 ∪ · · · ∪ T p i r ∪ { p i r +1 , . . . , p i j } of G whose associated homological sequence is B 0 = (n0 , n1 , . . . , n j ). Now, we check if the number of connected components must increase or decrease and then, we extend f : • If n j +1 − n j = 1 we take a 0-simplex p j +1 ∈ V c not in T j verifying that for some critical 0-simplex p it in T j there is a unique 1-simplex in E c contained in the unique path joining p it and p j +1 . We set f ( p j +1 ) = max f ( p 0 ), . . . , f ( p j ) + 1. Now, we define f on T p j+1 as in Step 1 and we take T j +1 = T j ∪ T p j+1 . • If n j +1 − n j = −1 we take a 1-simplex p j +1 = uv in E c not in T j such that there exist two critical 0-simplices in T j joined by a path in G including p j +1 . We set f ( p j +1 ) = max f ( p 0 ), . . . , f ( p j ), f (u ), f ( v ) + 1 and T j +1 = T j ∪ { p j +1 }. It is easy to check that f is an excellent discrete Morse function on T j +1 whose homological sequence is B 0 = (n0 , n1 , . . . , n j , n j +1 ). In the last step, we need to consider a 1-simplex in E c since n2k = 1 and n2k−1 must be 2 (n2k − n2k−1 = −1). Notice that in this step the function is defined on the whole of G. At the end of this construction we obtain an excellent discrete Morse function on G whose sequence B 0 is n0 , n1 , . . . , n2k . 2. Let us now consider that G is the union of b1 copies of S 1 and r trees T 1 , . . . , T r , that is, G= b1 S1 ∪ T1 ∪ · · · ∪ Tr. In the homological sequences of an excellent discrete Morse function on G, we can see that there exist exactly b1 values of t such that B 1 (t + 1) − B 1 (t ) = 1. By Lemma 3.5, for these values of t we have B 0 (t ) = B 0 (t + 1). Thus, the homological sequences B 0 and B 1 obtained in this case are B 0 : n0 , . . . , nt1 , nt1 , nt1 +1 , . . . , B1: 0, . . . , 0, 1, 1, ntb , 1 ntb , ntb +1 , . . . , n2k , 1 1 . . . , b 1 − 1, b 1 , b1 , (2) . . . , b1 . To count these sequences, we remove the copies of nt i for i = 1, . . . , b1 in the sequence B 0 and we obtain a walk of length 2k like in the case of trees (there are C k different such walks). On the other hand, the sequence B 1 is determined m−1 m−by  = 2k1 the position of the b1 1-simplices which are added to complete the copies of S 1 . Therefore, there are C bm−1 = b 1 1 different sequences since the first critical element must be a 0-simplex, and the number of homology classes  m−equivalence 1 in this case is less than or equal to the number of different pairs of sequences B 0 and B 1 , namely C k 2k . In order to prove the equality, given sequences B 0 and B 1 like in Eq. (2), we can construct an excellent discrete Morse function f on G with these homological sequences. Again, we begin subdividing G to have enough simplices. Let us choose the m simplices which will be the critical elements of the Morse function in this way: we select the 0-simplex p which joins the copies of S 1 , one 1-simplex e i for each S 1 and the remaining selected simplices in one of the trees, for example T 1 (again, by subdividing sufficiently many times if necessary, we can get that the number of simplices of T 1 is greater than 2k). As we have seen before, we can construct an excellent discrete Morse function g on the tree T = G − {e 1 , e 2 , . . . , eb1 } whose sequence B 0 is n0 , n1 , . . . , nt1 , nt1 +1 , . . . , n2k 3098 R. Ayala et al. / Topology and its Applications 156 (2009) 3091–3100 and we can suppose that g has its global minimum on p. Moreover, in order to get this function, we have previously decomposed T in k + 1 trees rooted in 0-simplices q j and k 1-simplices ql , where q0 , . . . , q2k are the critical elements of g with critical values c j = g (q j ) for i = 0, . . . , 2k. Next, starting from g, let us construct a new excellent function f on G having the given homological sequences. The critical elements of f are those of g, q0 , . . . , q2k , together with the 1-simplices e 1 , . . . , eb1 , where every edge e i must be between in qt i and qt i+1 , that is, f (qt i ) < f (e i ) < f (qt i+1 ). First, let us set f = g on the forest F 1 generated by q0 , q1 , . . . , qt1 . So, the first t 1 + 1 critical elements of f and g are the same, that is, p j = q j for j  t 1 . In order to obtain B 1 (t 1 + 1) = 1 at this step, we need to complete a copy of S 1 . So the next critical element of f must be e 1 . Therefore, we set pt1 +1 = e 1 and we define f (e 1 ) = max{ f (u 1 ), f ( v 1 ), ct1 } + 1. Notice that e 1 = u 1 v 1 and since u 1 and v 1 belong to F 1 , f is already defined on them. In the forest F 2 generated by qt1 +1 , . . . , qt2 , we set f = g + C 1 where C 1 = f (e 1 ) − ct1 +1 + 1. Now, the new critical elements of f are p j = q j −1 for t 1 + 1  j  t 2 . So we get that the critical values of f will be different. In fact, we have f (qt1 ) < f (e 1 ) < f (qt1 +1 ) since f (e 1 ) > ct1 = f (qt1 ) and   f (qt1 +1 ) − f (e 1 ) = ct1 +1 + f (e 1 ) − ct1 +1 + 1 − f (e 1 ) = 1 > 0. In a similar way, we define f as follows: f (e i ) = max{ f (u i ), f ( v i ), f (qt i )} + 1 where e i = u i v i and f = g + C i on the forest F i generated by qt i−1 +1 , . . . , qt i being C i a suitably chosen constant to assure that the critical values are different. Besides obtaining different critical values, we have the following relations between the level subcomplexes of the tree T = G − {e 1 , e 2 , . . . , eb1 } and G: G (a j ) = T (c j ), for 0  j  t 1 , G (a j ) = T (c j −r ) ∪ {e 1 , . . . , er }, for t r + 1  j  t r +1 , 1  r  b1 − 1 and G (a j ) = T (c j −b1 ) ∪ {e 1 , . . . , eb1 }, for tb1 + 1  j  m, where a j = f ( p j ) are the critical values of f . Therefore, we obtain an excellent discrete Morse function f on G whose homological sequences are the given ones. In consequence, the number of homology equivalence classes of excellent discrete Morse function for this type of graphs  m−1 is C k 2k . 3. If G is the wedge of some copies of S 1 , then the homological sequences satisfy B 0 (m − 1) = B 0 (m) = 1, B 1 (m) = b1 and B 1 (m) − B 1 (m − 1) = 1. That is, every excellent discrete Morse function on G reaches its global maximum on a critical 1-simplex e, which completes one of the copies of S 1 . If we remove e from G, we obtain the union of the wedge of copies of S 1 and trees or just a tree if G = S 1 . Notice that two excellent discrete Morse functions f and g on G are homologically equivalent if and only if their restrictions to G − {e } are homologically equivalent. Then, the number of homology equivalence classes of excellent discrete Morse functions with m critical elements on G is equal to the number of these equivalence classes for m − 1 critical elements 2k m−2 and G − {e }. Thus, if b1 = 1 we have m = 2k + 2, and we obtain C k = C k 2k = C k 2k equivalence classes, since G − {e } is a tree. Besides, if b1 > 1 we obtain C k m−2 2k = Ck (m−1)−1 2k equivalence classes. ✷ In the following example we clarify the constructions described in the above theorem: Example 6.2. Let us define an excellent discrete Morse function on the graph G of the figure below where dotted lines are rays: with 8 critical elements and whose homological sequences are B0 : 1 , 2 , 3 , 2 , 2 , 3 , 2 , 1 , B1 : 0, 0, 0, 0, 1, 1, 1, 1. R. Ayala et al. / Topology and its Applications 156 (2009) 3091–3100 3099 Let C be the unique cycle in G and let I be the infinite tree such that C ∩ I is a unique vertex p 0 . We begin selecting the critical elements: we take p 0 , the edge e in S 1 and the 6 simplices of I shown in dark in the picture below: First, we can define an excellent discrete Morse function g on the tree T = G − {e } in several steps. In the first step, we define the Morse function g on T p 0 by levels: Then, we define g on T p 1 ∪ T p 2 : Next, we define g on one selected 1-simplex uv between two 0-simplices already considered using increasing critical values and setting f (uv ) > max{ f (u ), f ( v )}: In the following steps we define g on a tree or a 1-simplex depending on whether B 0 (i + 1) − B 0 (i ) = 1 or −1: 3100 R. Ayala et al. / Topology and its Applications 156 (2009) 3091–3100 On the forest F generated by the first 4 critical elements of g, we set f = g and we assign to e a value greater than the values of f on its vertices and on the last critical element: Finally on G − ( F ∪ {e }) we set f = g + C where C = f (e ) − 4 + 1 = 2: As we can see in the last picture, the excellent Morse function f has the given homological sequences. Acknowledgement We would like to thank to the referee for his valuable comments and suggestions which led to improvements in this paper. 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