, provides foundations for tools in both computational topology and topological data analysis. This ubiquity comes from its combinatorial nature that translates to algorithms in a straightforward manner. In particular its core structure, called discrete gradient vector field, is defined through combinatorial matching, which allows to leverage a large literature of efficient algorithms.
This combinatorial nature lead to several applications of topological data analysis, where a common goal is to analyse the dynamics of a scalar function defined on the vertices of a cell complex . For such setup, greedy matching have been widely used to construct a discrete gradient field on out of [3, 11, 15, 10, 19]111Non-greedy matching approaches have also been devised, for example fusing progressive simplification  or locally enforced properties [19, 1].. The general principle of those greedy approaches relies on matching adjacent cells, weighing valid pairs aligned with the direction of steepest descent of . This leads to efficient, scalable and generalizable algorithms.
However, while Forman’s theory ensures the topological coherence of the resulting discrete gradient field, the later is not always guaranteed to be faithful to : its critical elements are not necessarily facets incident to piecewise-linear critical vertices, as defined by Banchoff , even in the simplicial case. For specific cases, when the weights guiding the greedy matching are given by a lexicographic ordering 
or a linear interpolation of,proofs of the faithfulness of the construction have been given. However those proofs are very intricate and require initial barycentric subdivisions [3, 14] to ensure enough regularity of the sampling of the function .
This work pins down the notion of regularity needed by greedy constructions, which we call discrete smoothness. This leads to simpler and broader proofs of the faithfulness of a greedy discrete gradient field construction. Moreover, we show that barycentric subdivisions ensures discrete smoothness, the proposed result generalizes previous ones [3, 14].
More precisely, given a simplicial complex and a scalar function sampled at its vertices, we introduce the notion of discrete smoothness of the pair , as a minimal sampling condition to ensure that a greedy discrete gradient field can be faithful to . The condition is necessary and sufficient for the critical elements to be well positioned by the greedy construction.
We structure the following study around a greedy algorithm where weights of matching adjacent cells are taken from lexicographic ordering of their vertices according to , following Babson and Hersh’s work . When applied on a smooth pair of any finite dimension, this construction leads to isolated critical elements, similarly to smooth Morse functions.
2 Greedy Matching
Most concepts of graph theory and their notations have been derived from .
is an ordered pairconsisting of a set of vertices and a set of edges disjoint of . Each edge of is associated with an unordered pair of vertices of . For , an edge is denoted by or without distinction. We say that and are extremes of the edge and that is incident to vertices and .
A path in a graph is a sequence of vertices denoted by such that from each of its vertices there is an edge to the next vertex in the sequence. In addition, the edges of a graph may have an associated weight. In a weighted graph, every edge is associated with a real and finite number which we will call weight. We will define or as the weight associated with the edge .
A graph is a subgraph of denoted by if and . We say that a subgraph is an induced subgraph of and denoted by , where is a subset of vertices of , if for any pair of vertices and of , is an edge of if and only if is an edge of . This means that is an induced subgraph of if it has all the edges that are in on the same set of vertices.
A matching in a graph is a subset of edges such that every vertex is contained in at most one edge of . Edges in are called matched edges and the extreme vertices to the matching edges are called saturated vertices. Similarly, edges that are not in are said to be unmatched and the vertices where all the edges are not matched are called unsaturated vertices.
An alternating path in a matching , defined in a graph , is a path whose edges alternate between those that are in and those that are not in . can therefore be defined as where is the number of edges in the path, that is, its length. Therefore, if then for all .
A digraph is an ordered pair consisting of a non-empty set of nodes and a set of arcs disjoint of . At each arc of an ordered pair of nodes is associated. A digraph is nothing more than a directed graph; the pairs of nodes that make up an arc are now ordered. For , an arc is denoted by and implies that is directed from to . As in graphs, a digraph can be weighted, with all its arcs associated with a finite and real number.
The main algorithm from matching theory that we will use is simply a greedy matching in graphs. We rewrite its steps in pseudocode form as shown in the next algorithm.
At each iteration, the algorithm selects the edges with the lowest weights in the graph by creating the set of minimum edges , adds them to the matching and finally removes the vertices that are extreme from these edges and, therefore, their incident edges. Thus, at each iteration an induced subgraph of is created. These steps of selecting edges, matching, and removing vertices repeat until all edges in have been removed.
At this moment it is worth noting thatis only a matching if the intersecting edges (neighboring edges) of the graph are tie free. This means that in there is no edge adjacent to edge such that . It is fundamental to understand this sufficient condition because it will be used throughout the paper.
We now propose and define one notation that will be useful in order to simplify the lemmas and proofs related to the theoretical guarantees of this algorithm.
First, it is necessary to remember that we denote the as function evaluated on an edge as the weight of that edge . We are ratifying this because we will define another function, also named as , but defined in the vertices of the graph induced by a matching .
Thus, for a matching in , the function is such that
We will call the function therefore by saturation function. The saturation function is well defined because is a matching, so each vertex can only be saturated by at most one edge. In addition, the matching type will always be implied in context.
Now we will define as the set of vertices of whose saturation function is greater than or equal to the real number . Then consider as the induced subgraph of for the set of vertices . This induced subgraph contains only the vertices whose saturation function is greater than or equal to the real number . We denote therefore the set of vertices and the set of edges of as and , respectively.
By definition, is a subgraph of where the minimum weight of its edges is . This notation is interesting because it identifies the subgraph of at the moment that the edges of a certain weight are selected in the matching step of the algorithm. In other words, we can recognize the iteration in which the algorithm is executing when we refer to .
With this in mind, it is also easy to note that if a weight edge exists in the graph then surely that edge is matched in . The next lemma formalizes this idea.
Let be a greedy matching in a graph . If , then .
The greedy matching has some properties in relation to its set of edges . These properties are strongly related to a particular type of alternating path and will be useful in assisting various proofs on the discrete gradient further along. For that, we need to define some particular cases of alternating paths. Here is the definition of what we call the maximal alternating path (see examples in Figure 1).
[Maximal Alternating Path] We denote as maximal alternating path, any alternating path such that if a vertex is saturated, then it is saturated by an edge in .
An important result of the greedy algorithm is the notion of occurrence of edges of minimum weight on maximal alternating paths. By means of the Theorem 2, we can say that for any maximal alternating path, the edges of minimum weight will be matched.
If is a maximal alternating path, then .
It is important to note that a maximal alternating path such that and its vertices are unsaturated, deserves some attention. Note that this type of construction never happens in a greedy matching. Although not so intuitive, note that this is verifiable by the last theorem.
3 Preliminaries for the Discrete Gradient Field
A finite simplicial complex is a set of vertices together with a set of subsets of , such that must satisfy some properties:
if and , then
We will refer to the simplicial complex as . The elements of are called simplexes. A simplex is said of dimension , denoted by , if contains vertices.
A simplex is also said to be a facet of another simplex, denoted by , if and . We finally say that the dimension of is the largest dimension of these simplexes.
Another very common way to represent a simplicial complex is in terms of a digraph. This structure is called the Hasse diagram and is defined below .
[Hasse Diagram] Given a simplicial complex , its Hasse diagram is the directed graph where the set of nodes is the set of simplexes of and the set of arcs in is composed of all pairs such that if and only if .
The Hasse diagram, therefore, acts as a kind of dictionary between the concepts of graphs and simplicial complexes. It is through its structure that we can understand how the discrete gradient is constructed.
[Discrete Vector Field] A discrete vector field in is a collection of simplexes pairs in with , such that each simplex is at most a pair of .
We write if , that is, if the arc is matched, and otherwise.
Unsaturated nodes and alternate paths in the Hasse diagram are also defined in a particular way.
[Critical Simplex] A simplex is said critical for a discrete vector field if it does not belong to any pair of .
 Given a discrete vector field in a simplicial complex , a is a sequence of simplexes such that for we have that and .
A is then an alternating path in the Hasse diagram with the constraint that all dimensions of the simplexes alternate between or , with .
If a path is such that and , we call it non-trivial closed. We say that is a discrete gradient field if it does not contain any non-trivial closed.
[Discrete Gradient Field] A discrete vector field is a discrete gradient field if and only it does not contain non-trivial closed .
So, given the simplicial complex associated with a discrete object, the construction of the gradient is given by a matching in its Hasse diagram.
4 Greedy Discrete Gradient Field
As we mentioned in the introduction, we would use the discrete gradient as a way to study a function evaluated at the vertices of a discrete object represented by . We also mention that there are several algorithms that propose methods of construction of the discrete gradient field.
The vast majority of those algorithms are summed up in greedy matchings in weighted digraphs together with a choice of inducing weights of to the arcs of . There are several ways to induce the weight of the function to the arcs (and thereby make a weighted digraph) [12, 11, 10, 17, 16, 14]. In this work, we are following the choice of induction proposed in [16, 14].
We need a total order on the vertices of the simplicial complex. Given a injective function defined at the vertices of the simplicial complex. Then, we define as the set of weights associated with the vertices of , that is, .
We can extend the total order of the vertices to a total order of the simplicies with a lexicographic order.
[Lexicographic Order] If , then . If , then , where represents the subtraction operation in sets and represents the symmetric difference.
With this total order on the simplicies, we can now define the arc weights of the Hasse diagram.
[Arc Weights] Given a function defined at the vertices of a simplicial complex, the weight induced by it in the arcs of its Hasse diagram is given by .
Observe that the weight of the arc , for simplicial complexes, is the value of the function at the vertex . Since it is a simplicial complex, is facet of . Thus, the operation always results in a vertex where and . Therefore we have that .
By an abuse of notation, we sometimes drop , when the context is clear.
Once we have defined how to assign weights to the Hasse diagram arcs through a function evaluated at the vertices of a discrete object and how the greedy matching works, we can state in more detail how the discrete gradient is constructed (Algorithm 2).
5 Geometric Faithfulness of Greedy Discrete Gradient Field
First we will explain why the discrete vector field associated with the greedy matching is in fact a discrete gradient field (which is equivalent to the smooth gradient in the sense of not having closed orbits). Next we will enunciate another geometric notion for the same: a notion of decreasing flow. The first result is already known in the literature [16, 14] for the greedy algorithm, in particular, and other constructs; we just simplify the proof. The second is a new result. It is also worth mentioning that all the results are valid for any dimension.
To show that the discrete vector field associated with the greedy matching is in fact a discrete gradient field, it is the same as showing that such algorithm does not generate non-trivial closed (Definition 3). For this, we will propose some notations that will make the proof simple. The following notations and lemmas are our contributions to simplify evidence in the literature.
Considering a denoted by
we can write it as an alternating path
In this way, according to the indices, we have that and .
The discrete vector field associated with the greedy matching in the Hasse diagram is a discrete gradient field.
Following the theoretical assurances, the next theorem to be enunciated guarantees a notion of decreasing flow in a . Essentially, we want to show that from any simplex, by following the to the critical simplex, then the latter is smaller than any other simplex in the path.
In a discrete vector field associated with the greedy matching , for any we have that, if is critical, then for all .
Still in the Theorem 5, it is worth mentioning that if was not critical, then the path would not be a maximal alternating path (Definition 1). That way, we would not been able to use Theorem 2 in our proof.
There is a guarantee that the critical simplex where ends is the smallest of all simplexes in the path, however, we can not say the same for any intermediate path of simplexes in that .
Finally, to end this section, we will prove a sufficient condition for a simplex to be matched. First we define a set whose elements are vertices which provide the weights of all the arcs incident to .
or and .
The following lemma give a sufficient condition for a simplex to be matched.
[Steepest Descent] If there exists such that , then .
A straighfoward corollary is a necessary condition for a critical simplex.
If is critical, then and .
6 Discrete Smoothness
In this section, we introduce a concept of discrete smoothness for functions on a simplicial complex and then prove that for discrete smooth functions, the converse of the Steepest Descent Theorem (Theorem 5) holds (Theorem 2). In that case, we have a full characterization of the greedy Morse matching and its critical simplices. In Theorems 6.1 and 6.1, we show that the dynamics around critical simplices are similar to their counterparts from the smooth theory.
Our results on discrete smoothness generalize several results stated with barycentric subdivision [3, 13, 14], and we prove at the end of the Section 6 that any function defined on a simplicial complex is discrete smooth after one barycentric subdivision.
We begin with the main definitions of this section.
[Discrete Smoothness] A simplex is discrete smooth, if and only if, if , then for all . A function is called discrete smooth on , if for every , is discrete smooth.
Note that this definition depends on the function as well as the structure of the simplicial complex , and we write that the simplicial complex is discrete smooth when the function defined on is discrete smooth. In this entire section we assume that is discrete smooth.
Observe that the simplicial complexes in the right of Figure 2 are not discrete smooth.
With discrete smoothness we have a full characterization of the greedy Morse matching.
[Steepest Descent with Discrete Smoothness]
If is smooth, then such that .
This characterization gives an important and extremely fast, linear and naturally parallel algorithm to compute greedy Morse matching (in the discrete smooth case): the algorithm only needs to check every and to see if is matched above.
With the converse of the Steepest Descent Theorem, we can now prove the converse of Corollary 5, which follows directly by applying the converse of the Steepest Descent Theorem. is critical if and only if and .
6.1 Geometrically Faithful Critical Simplicies and Paths
Now that we have characterized the critical simplices, we want to understand the behavior of the greedy Morse matching in their neighborhood. We show that this behavior is similar to the local dynamics around the continuous critical points. In this entire section we assume that is discrete smooth.
If is critical, then all facets of are matched above.
If is critical, then for all .
The main characterization of the critical simplicies and its neighborhood follows easily from the two above Theorem.
[Geometrically Faithful Critical Simplicies] is critical if and only if for all and all facets of are matched above.
Figure 3 shows the Theorem above on surfaces (2-manifolds). In particular, Theorem 6.1 states that all edges adjacent to a minimum points towards the minimum, and Theorem 6.1 states that all facets of a maximum points outwards that maximum.
When the complex is smooth, we obtained a much stronger result than Theorem 5, where we have truly decreasing paths. In a discrete vector field associated with the greedy matching , for any we have that, if is critical, then .
6.2 Discrete Smoothness of the Barycentric Subdivision
The goal of this section is to prove that, for any function defined on a simplicial complex , a function induced by on the barycentric subdivision of is discrete smooth. This proves that the results of the previous sections apply to barycentric subdivision, generalizing previous results on similar greedy constructions of Morse matchings [3, 13, 14].
[Barycentric Subdivision] The barycentric subdivision of a simplicial complex is a simplicial complex constructed as follows:
for every simplex , there is a vertex ,
for every sequence , such that , there is a -simplex .
Since each vertex of the subdivision corresponds to an original simplex of , we extend the function to the vertices in ordering vertex according to the lexicographic ordering of the vertices of with a function (see Figure 4).
[Extension of ] The extension of function on the vertices of is defined by . The values of are totally ordered using lexicographic order from Definition 4. Since is a simplicial complex and is injective, then is injective.
To illustrate the ordering, consider as a single triangle and its faces, with . The lexicographic ordering orders the faces of as: .
[Discrete Smoothness of the barycentric subdivision] Given a function on , the extension of the function on the barycentric subvision of is smooth.
7 Greedy Matchings on Polyhedral Complexes
In this section, we extend the greedy matchings beyond simplicial complexes.
A polyhedral complex in is a set, , consisting of a (finite) set of convex polytopes in satisfying the following conditions:
Every face of a polytope in also belongs to .
For any two polytopes and , if , then is a common face of both and .
To deal with polyhedral complexes, we define a modified Hasse diagram where the function is taken under consideration.
[Modified Hasse Diagram] Given a simplicial complex , its Hasse diagram is the directed graph where the set of nodes is the set of simplexes of and the set of arcs in is composed of all pairs such that if and only if and .
Observe that for simplicial complexes, this definition does not reduces to Definition 3. This is not in fact a generalization of previous sections but an alternative when dealing with polyhedral complexes. Please note that, since arcs of the original Hasse diagram are missing, there might be two critical adjacent cells, which was impossible in the previous definitions in simplicial complex.
If we consider this modified Hasse diagram, we can show that the greedy matching algorithm constructs a discrete gradient field with decreasing paths such as Theorem 6.1 and its follows that the matching produced is in fact a discrete gradient field.
In a discrete vector field associated with a matching in the Modified Hasse diagram , for any we have that .
The discrete vector field associated with a matching in the Modified Hasse diagram is a discrete gradient field.
8 Application to CAT(0) Cubical Complexes
In this section, we will show that CAT(0) cubical complexes are collapsible (a result established in  using convexity), by applying the greedy matching on the modified Hasse diagram from the previous section. We will use a fully combinatorial description of the CAT(0) cubical complex recently developed in . We include their combinatorial description verbatim from  for completeness. Please see the reference for examples and details.
Recall that a poset is locally finite if every interval is finite, and it has finite width if every antichain (set of pairwise incomparable elements) is finite.
 A poset with inconsistent pairs is a locally finite poset of finite width, together with a collection of inconsistent pairs , such that:
If and are inconsistent, then there is no such that and .
If and are inconsistent and , then and are inconsistent.
In particular, notice that any two inconsistent elements must be incomparable. A diagram of a poset with inconsistent pairs is obtained by drawing the poset, and connecting each minimal inconsistent pair with a dotted line. An inconsistent pair is minimal if there is no other inconsistent pair with and (See Figure 5).
 If is a poset with inconsistent pairs, we construct the cube complex of , which we denote . The vertices of are identified with the consistent order ideals of . There will be a cube for each pair of a consistent order ideal and a subset , where is the set of maximal elements of . This cube has dimension , and its vertices are obtained by removing from the possible subsets of . These cubes are naturally glued along their faces according to their labels.
[Combinatorial description of CAT(0) cubical complexes.]  There is a bijection between posets with inconsistent pairs and rooted CAT(0) cube complexes, given by the map .
The next theorem was already proven in  but here we give fully combinatorial proof. We will define a order on the cubes of , construct a discrete gradient field with the greedy matching on the modified Hasse diagram, and finally we will show that there is only one critical vertex in the discrete gradient field. Therefore the complex is collapsible.
CAT(0) cube complexes are collapsible.
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Appendix A Appendix
a.1 Proof of Theorem 2
If is a subgraph of such that we have , then is a subgraph of .
If is a subgraph of , then and . In addition, if we have , then . Thus, . Now consider an edge . It is easy to verify that and . This means that and . Since is an induced subgraph of for , it follows that , and therefore . Therefore, is a subgraph of . ∎
Let . If is a maximal alternating path, then we have .
Consider as one of the edges of minimum weight in . If is a maximal alternating path, then we have that is saturated by an edge of or is unsaturated (Definition 1). Thus, follows that or , respectively. ∎
Proof of Theorem 2
As and are facets of in the , we will call as and two vertices. We can simplify those equalities checking that and . This indicates that, roughly speaking, for any intermediate defined as , we have that obtained a vertex and lost a vertex in relation to the previous simplex . Thus, we will name as and two sets of vertices. The first one represents a set of vertices that was obtained throughout the path and the second one is a set of vertices that was lost throughout the path.
This means that, is the set of weights of the matched edges and is the set of weights of the unmatched edges in a .
The relations of those sets, and , with the simplexes along a are intuitive and consist of useful tools to prove results related with the discrete gradient field.
As the proof is symmetrical, consider only one of the relations. If a vertex then and . Consider the two next statements.
If then there are at least a pair of simplexes and for in such that . This implies that and .
Similarly, if so there is no pair of simplexes and for in such that . This means that if .
Combining the two statements, we conclude that and . ∎
For a non-trivial closed , we have that .
If a denoted by is non-trivial closed, then and . Thus, we have that and . This means that
The corollary shows that the set of all the lost vertices and the set of all the obtained vertices in a non-trivial and closed are the same. This is not a surprise since .
Suppose by contradiction that is not a discrete gradient field. Therefore, by the Definition 3, there exists a closed non-trivial denoted by
where . Because is an alternating cycle, it is also a maximal alternating path, then (Theorem 2). Since is the set of weights of the matched edges in and is the set of weights of the unmatched edges in , we have that which also belongs to (Lemma A.2). As , then . Contradiction. ∎
If a vertex then and . This means that exist a pair of simplexes and where such that and . Thus, . Similarly, if a vertex then and . This means that exist a pair of simplexes and where such that and . Thus, .
This last lemma simply indicates that if a vertex is not in the initial simplex of the but it is in the final simplex, then this vertex is in the set of obtained vertices. Similarly, if a vertex is in the initial simplex but is not in the final simplex, then this vertex is in the set of lost vertices.
Consider a denoted by
with critical. Because is critical, then is a maximal alternating path, then (Theorem 2). Since is the set of weights of the matched edges in and is the set of weights of the unmatched edges in , we have that which also belongs to (Lemma A.2). As (Lemma A.2, then , this means that . ∎
a.3 Proof of Theorem 5
If , then and . Also .
Trivial, just set operations. ∎
Proof of Theorem 2
Proof of Theorem 6.1
Proof of Theorem 6.1
a.5 Proof of Theorem 4
If , then let and .
Let . Since , then . It is easy to see that . Therefore and it follows that . In addition, and . Therefore, by the lex order, . ∎
Proof of Theorem 4
Let and where , for some . To prove discrete smoothness, we must show that . We will first show that for all (1), (2), and (3):
Now suppose , then . Therefore . Then . Since , it is easy to see that , and . We know that . Finally we have . ∎
Proof of Theorem 7
Proof of Theorem 7
Suppose by contradiction that is not a discrete gradient field. Therefore, by the Definition 3, there exists a closed non-trivial denoted by where . As the sequence of simplexes are strictly decreasing, represents a contradiction. ∎
a.7 Proof of Theorem 8
Let be a consistent order ideal and let . The faces of the cube in the cubical complex are the cubes , where and are disjoint subsets of N. The maximal cubes in correspond to the maximal consistent antichains of .
We can order the vertices of , which by Definition 5 are the consistent order ideals of , with any linear ordering of the elements of . Now we can use a shortlex ordering to order all the cubes such as: if and only if or and . Note the opposite signs when comparing and or and .
In shortlex ordering, the sequences are primarily sorted by cardinality (length) with the shortest sequences first, and sequences of the same length are sorted into lexicographical order from Definition 4.
Proof of Theorem 8
Apply the greedy matching with the cubes ordered by the order above. If , then define . We will show that if , then (See Figure 5). With this we can conclude that every is matched except when , since if , then , if or if . Therefore is collapsible since it has one critical vertex, the empty set ideal.
Let and .
Since , then . Therefore . Therefore , by Definition 7.
Suppose . Since , then for some . We have that .
Suppose . Since , then or for some . We have that or , since .
Suppose . Since , then for some . We have that