aa r X i v : . [ m a t h . A T ] J u l A Discrete Morse Theory for Digraphs
Chong Wang* † , Shiquan Ren* Abstract . Digraphs are generalizations of graphs in which each edge isassigned with a direction or two directions. In this paper, we define discreteMorse functions on digraphs, and prove that the homology of the Morsecomplex and the path homology are isomorphic for a transitive digraph. Wealso study the collapses defined by discrete gradient vector fields. Let G bea digraph and f a discrete Morse function. Assume the out-degree and in-degree of any zero-point of f on G are both 1. We prove that the originaldigraph G and its M -collapse ˜ G have the same path homology groups. Digraph is an important topology model of complex networks, and its topological structurehas important research value and wide application prospect in data science research. Forexample, J. Bang-Jensen and G.Z. Gutin [5] studied digraphs and gave applications ofdigraphs in quantum mechanics, finite automata, deadlocks of computer processes, etc.The path homology of digraphs was first defined and studied by A. Grigor’yan, Y. Lin,Y. Muranov and S.T. Yau in [7] in 2012. Subsequently, in 2014, it is proved in [12] thatthe path homology is functorial with respect to morphisms of digraphs, and is invariantup to certain homotopy relations of these morphisms. In 2015, A. Grigor’yan, Y. Lin, Y.Muranov and S.T. Yau [8, 9] studied the cohomology of digraphs and graphs by using thepath homology theory. Moreover, in 2017, A. Grigor’yan, Y. Muranov, and S.T. Yau [11]proved some Künneth formulas for the path homology with coefficients in a field. In 2018,A. Grigor’yan, Y. Muranov, V. Vershinin and S.T. Yau [10] generalized the path homologytheory of digraphs and constructed the path homology theory of multigraphs and quivers.Discrete Morse theory originated from the study of homology groups and cell structure ofcell complexes. By reducing the number of cells and simplices, discrete Morse theory is help-ful to simplify the calculation of homology groups. Moreover, discrete Morse theory providestheoretical supports for the calculation of persistent homology in the field of topological dataanalysis [19, 20, 21]. In 1998, R. Forman [13] invented the discrete Morse theory for simpli-cial complexes or general cell complexes, as a discrete version of the classical Morse theoryof smooth Morse functions. In the subsequent references [14, 15, 16], R. Forman studiedthe discrete Morse theory, the cup product of cohomology, and Witten Morse theory based
Primary 55U15 55N35; Secondary 55U35.
Keywords and Phrases. digraphs, Morse function, discrete gradient vector fields, acyclic matching, M -collapse.* first authors; † corresponding author.
1n [13]. In 2005, Dmitry N. Kozlov [17] extended the combinatorial Morse complex con-struction to arbitrary free chain complexes, and give a short, self-contained, and elementaryproof of the quasi-isomorphism between the original chain complex and its Morse complex.From 2007 to 2009, R. Ayala et al. [1, 2, 3, 4] studied the discrete Morse theory on graphsby using the discrete Morse theory of cell complexes and simplicial complexes given by R.Forman.Let R be an arbitrary commutative ring with a unit. Let G = ( V, E ) be a digraph. An n -path is a sequence v v · · · v n of n + 1 vertices in V where for each ≤ i ≤ n , v i − and v i are assumed to be distinct. Let Λ n ( V ) be the free R -module consisting of all the formallinear combinations of the n -paths on V . Let ∂ n = P ni =0 ( − i d i . Then ∂ n is an R -linearmap from Λ n ( V ) to Λ n − ( V ) satisfying ∂ n ∂ n +1 = 0 for each n ≥ (cf. [7, 12, 9, 11]). Hence { Λ n ( V ) , ∂ n } n ≥ is a chain complex. An allowed elementary n -path is a n -path such thatfor each ≤ i ≤ n , v i − → v i is a direct edge of G . For simplicity, we assume that in anallowed elementary n -path, v i − = v i for each ≤ i ≤ n . Let P n ( G ) be the free R -moduleconsisting of all the formal linear combinations of allowed elementary n -paths on G . Then P n ( G ) is a sub- R -module of Λ n ( V ) . Note that under the boundary operator ∂ , the imageof an allowed elementary path does not have to be allowed. Hence ∂ may not map P n ( G ) to P n − ( G ) . Nevertheless, P n ( G ) has the sub- R -module Ω n ( G ) which which consists of allthe ∂ -invariant elements in P n ( G ) . We define the path homology of G as the homology ofchain complex { Ω n ( G ) , ∂ n } .In this paper, we define the discrete Morse functions on digraphs in Section 3, andanalysis the basic properties in Section 4. In Section 5, we study the discrete gradientvector fields and Morse complexes of digraphs. Let G be a transitive digraph. Then we givethe first main result of this paper in the following theorem. Theorem 1.1.
Let G be a transitive digraph. Let R be an arbitrary commutative ring witha unit. Then the chain complex Ω ∗ ( G ) can be decomposed as a direct sum of the Morsecomplex and atom chain complexes: Ω ∗ ( G ) = C M∗ ( B ∗ ( G )) M (direct sum of atom chain complexes) . (1.1) And for each n ≥ , the path homology of G is isomorphic to the homology of the Morsecomplex H n ( { Ω ∗ ( G ) , ∂ ∗ } ; R ) ∼ = H n ( { C M∗ ( B ∗ ( G )) , ∂ M∗ } ; R ) (1.2) where the explicit formula of ∂ M∗ is given in Definition 8. Let f : V ( G ) −→ [0 , + ∞ ) be a discrete Morse function on G . We substitute the orderedtriple of vertices ( u, v, w ) satisfying f ( v ) = 0 , u → v → w and u → w in G with the orderedcouple of vertices ( u, w ) , and remove the directed edges u → v and v → w . The resultingdigraph ˜ G and the restriction of f on ˜ G (denoted by ˜ f ) form a pair ( ˜ G, ˜ f ) , which is called the2 -collapse of the pair ( G, f ) . In Section 6, we give the second main result in the followingtheorem. Theorem 1.2.
Let G be a digraph and f : V ( G ) −→ [0 , + ∞ ) a discrete Morse function on G . Assume the out-degree and in-degree of any zero-point of f on G are both 1. Then thenatural inclusion map i : ˜ G → G induces an isomorphism from the path homology groups of ˜ G to the path homology groups of G , where ˜ G is obtained by M -collapse of the pair ( G, f ) . Particularly, a graph is a digraph where each edge is assigned with two directions. Sinceall discrete Morse functions on graphs have strict positive values on all the vertices, the(combinatorial) discrete gradient vector fields are all empty. Hence, both of the two mainresults are trivial for graphs.
In this section, we mainly review some basic concepts and theorems in [5, 7, 12] and givesome properties of digraphs. A digraph G = ( V, E ) is a couple of a set V whose elements are called the vertices, and asubset E ⊂ { V × V \ diag } of ordered pairs of vertices whose elements are called directededges or arrows. The directed edge with starting point v and ending point w is denoted by v → w . Triangles and squares are simple examples of digraphs. A triangle is a sequenceof three distinct vertices v , v , v ∈ V such that v → v , v → v , v → v . A square is asequence of four distinct v , v , v ′ , v ∈ V ,such that v → v , v → v , v → v ′ , v ′ → v (cf.[7, Section 4.2]).We define Ω n ( G ) = P n ( G ) ∩ ∂ − n P n − ( G ) , Γ n ( G ) = P n ( G ) + ∂ n +1 P n +1 ( G ) . Then as graded R -modules, Ω ∗ ( G ) ⊆ P ∗ ( G ) ⊆ Γ ∗ ( G ) ⊆ Λ ∗ ( V ) . (2.1)And as chain complexes, { Ω n ( G ) , ∂ n | Ω n ( G ) } n ≥ ⊆ { Γ n ( G ) , ∂ n | Γ n ( G ) } n ≥ ⊆ { Λ n ( V ) , ∂ n } n ≥ . By [6, Proposition 2.4], the canonical inclusion ι : Ω n ( G ) −→ Γ n ( G ) , n ≥ of chain complexes induces an isomorphism between the homology groups ι ∗ : H m ( { Ω n ( G ) , ∂ n | Ω n ( G ) } n ≥ ) ∼ = −→ H m ( { Γ n ( G ) , ∂ n | Γ n ( G ) } n ≥ ) , m ≥ , path homology of G . The definition of path homology here is essentially consis-tent with [7, Definition 3.12]. Definition 1. [5, Section 2.3] A digraph G is called transitive , if for any two directed edges u → v and v → w of G , there is a directed edge u → w of G . Proposition 2.1.
Let G be a transitive digraph. Then for each n ≥ , we have that Ω n ( G ) = P n ( G ) .Proof. By (2.1), Ω n ( G ) ⊆ P n ( G ) . On the other hand, for any allowed elementary n -path α = v · · · v n , by Definition 1, we have that d i ( α ) = v · · · v i − ˆ v i v i +1 · · · v n , ≤ i ≤ n is still allowed in G . That is, ∂ n α is a linear combination of allowed elementary ( n − -paths.Hence, Ω n ( G ) ⊆ P n ( G ) . The proposition follows. Let G and G ′ be digraphs. A morphism (or digraph map ) is a map f : V ( G ) −→ V ( G ′ ) suchthat for any vertices u, v ∈ V ( G ) , if u → v is a directed edge of G , then either f ( u ) = f ( v ) or f ( u ) → f ( v ) is a directed edge of G ′ (cf. [12, Definition 2.2]). We denote such a morphismshortly as f : G −→ G ′ . A morphism f : G −→ G ′ is called an isomorphism if f is abijection from V ( G ) onto V ( G ′ ) , and the inverse of f is also a morphism.A line digraph I n is a digraph whose the set of vertices is { v , v , . . . , v n } and the setof directed edges contains exactly one of the directed edges v i → v i +1 , v i +1 → v i for each i = 0 , , . . . , n − , and no other directed edges (cf. [12, p. 632]). Note that a path is aspecial line digraph with all the directed edges v i → v i +1 .Let G = ( V ( G ) , E ( G )) and H = ( V ( H ) , E ( H )) be two digraphs. Define the Cartesianproduct G ⊡ H as a digraph with the set of vertices V ( G ) × V ( H ) and the set of directededges as follows: for any x, x ′ ∈ V ( G ) and any ( y, y ′ ) ∈ V ( H ) , we have ( x, y ) → ( x ′ , y ′ ) ifand only if either x = x ′ and y → y ′ , or x → x ′ and y = y ′ (cf. [12, Definition 2.3]). Definition 2. (cf. [12, Definition 3.1]) Two morphisms f, g : G −→ H are called homotopic and denoted as f ≃ g if there exists a line digraph I n with n ≥ and a morphism F : G ⊡ I n −→ H such that F | G ⊡ { } = f and F | G ⊡ { n } = g . We call F a homotopy .Two digraphs G and H are called homotopy equivalent if there exist morphisms f : G −→ H and g : H −→ G such that f ◦ g ≃ id H and g ◦ f ≃ id G . We shall write G ≃ H (cf. [12,Definition 3.2]). Definition 3. (cf. [12, Definition 3.4]) A retraction of G onto H is a digraph map r : G → H such that r | H = id H . A retraction r : G → H is called a deformation retraction if i ◦ r ≃ id G ,where i : H → G is the natural inclusion map.4 roposition 2.2. [12, Proposition 3.5] Let r : G → H be a deformation retraction. Then G ≃ H and the maps r and i are homotopy inverses of each other. Theorem 2.3. [12, Theorem 3.3] Let f ∼ = g : G → H be two homotopic digraph maps.Then these maps induce the identical homomorphisms of path homology groups of G and H .Consequently, if the digraphs G and H are homotopy equivalent, then their path homologygroups are isomorphic. Lemma 2.4.
Suppose α ( n +1) > γ ( n ) > β ( n − are allowed elementary paths on G . Thestarting points of α ( n +1) , γ ( n ) and β ( n − are the same, and the end points of them are alsothe same. Then either of the followings holds:(a). there exists an allowed elementary n -path γ ′ ( n ) = γ ( n ) such that α > γ ′ > β ;(b). β is obtained by removing two subsequent vertices v i → v i +1 in α where ≤ i ≤ n − .Proof. Without loss of generality, we write α = v v · · · v n +1 and β = v · · · b v i · · · b v j · · · v n +1 .Since α and β have the same start points and the same end points, we have ≤ i < j ≤ n . Case 1 . i < j − . Subcase 1.1 . γ = v · · · b v i · · · v j · · · v n +1 .Then we let γ ′ = v · · · v i · · · b v j · · · v n +1 . Since β is an allowed elementary path and i = j − , we have that v j − → v j +1 is a directed path in G . Hence γ ′ is an allowedelementary path. Subcase 1.2 . γ = v · · · v i · · · b v j · · · v n +1 .Then we let γ ′ = v · · · b v i · · · v j · · · v n +1 . Since β is an allowed elementary path and i + 1 = j , we have that v i − → v i +1 is a directed path in G . Hence γ ′ is an allowedelementary path.In both Subcase 1.1 and
Subcase 1.2 , we can see that γ ′ ( n ) = γ ( n ) and α > γ ′ > β . Case 2 . i = j − .Then β is obtained by removing two subsequent vertices v i → v i +1 in α and ≤ i ≤ n − .Summarising Case 1 and Case 2, the lemma follows. Lemma 2.5.
Suppose α ( n +1) > γ ( n ) > β ( n − are allowed elementary paths on G . Thestarting points of α ( n +1) , γ ( n ) and β ( n − are not all the same, or the end points of themare not all the same. Then either of the followings holds:(a). there exists an allowed elementary n -path γ ′ ( n ) = γ ( n ) such that α > γ ′ > β ;(b). β is obtained by removing v → v or v n → v n +1 in α .Proof. It is sufficient to consider the following two cases.5 ase 1 . α and γ have the same start points and the same end points. Let α = v · · · v n +1 , γ = v · · · b v i · · · v n +1 for some ≤ i ≤ n . Subcase 1.1 . β = b v · · · b v i · · · v n +1 .Then we let γ ′ = b v v · · · · · · v n +1 . Obviously, γ ′ is an allowed elementary path and α > γ ′ > β , γ ′ = γ . Subcase 1.2 . β = v · · · b v i · · · [ v n +1 .Then we let γ ′ = v v · · · [ v n +1 . Obviously, γ ′ is an allowed elementary path and α >γ ′ > β , γ ′ = γ . Subcase 1.3 . β = v · · · b v i · · · b v j · · · v n +1 for ≤ i = j ≤ n .Then α , γ and β have the same start points and the same end points. This is impossible.Summarizing Subcase 1.1 , Subcase 1.2 and
Subcase 1.3 , we can see that γ ′ ( n ) = γ ( n ) and α > γ ′ > β . Case 2 . The starting points of α and γ are different, or the end points of them aredifferent. Subcase 2.1 . α = v · · · v n +1 , γ = b v · · · v i · · · v n +1 and β = v · · · b v i · · · v n +1 for some ≤ i ≤ n + 1 . We separate Subcase 2.1 into the following subcases: • i = n + 1 .Then β = v · · · v n . Let γ ′ = v v · · · [ v n +1 . We have that γ ′ is an allowed elementarypath such that α > γ ′ > β , γ ′ = γ . • ≤ i < n + 1 .Then we let γ ′ = v · · · b v i · · · v n +1 . Since β is an allowed elementary path and v i − → v i +1 is a directed path in G , γ ′ is an allowed elementary path such that α > γ ′ > β and γ ′ = γ . • i = 1 .Then β is obtained by removing two subsequent vertices v → v in α . Subcase 2.2 . α = v · · · v n +1 , γ = v · · · v i · · · [ v n +1 and β = v v · · · b v i · · · v n for some ≤ i ≤ n . We separate Subcase 2.2 into the following subcases: • i = 0 .Then β = v · · · v n . Let γ ′ = b v v · · · v n +1 . Then γ ′ is an allowed elementary pathsuch that α > γ ′ > β , γ ′ = γ . • < i < n .Then we let γ ′ = v · · · b v i · · · v n +1 . Since β is an allowed elementary path and v i − → v i +1 is a directed path in G , γ ′ is an allowed elementary path such that α > γ ′ > β and γ ′ = γ . 6 i = n .Then β is obtained by removing two subsequent vertices v n → v n +1 in α .Summarising Case 1 and Case 2, the lemma follows. Proposition 2.6.
Suppose α ( n +1) > γ ( n ) > β ( n − are allowed elementary paths on G .Then either of the followings holds:(a). there exists an allowed elementary n -path γ ′ ( n ) = γ ( n ) such that α > γ ′ > β ;(b). β is obtained by removing two subsequent vertices v i → v i +1 in α where ≤ i ≤ n .Proof. By Lemma 2.4 and Lemma 2.5, the proposition follows directly.The following example shows that for any α ( n +1) > γ ( n ) > β ( n − , there may not existan allowed elementary path γ ′ ( n ) such that α ( n +1) > γ ′ ( n ) > β ( n − . Example 2.7.
Let V = { v , v , v , v } . Let G be a digraph with the set of vertices V andthe set of directed edges { v → v , v → v , v → v , v → v , v → v } . Let α (3) = v → v → v → v , γ (2) = v → v → v , and β (1) = v → v . v v v v We have that there is no allowed elementary -path γ ′ (2) = γ (2) such that α (3) > γ ′ (2) > β (1) ,where β (1) is obtained by removing two subsequent vertices v → v in α (3) . In this section, we define the discrete Morse functions on digraphs and the critical allowedelementary paths of the discrete Morse functions.
Definition 4.
A map f : V ( G ) −→ [0 , + ∞ ) is called a discrete Morse funtion on G , if forany allowed elementary path v v · · · v n on G , both of the followings hold:(i). there exists at most one v i ( ≤ i ≤ n ) with f ( v i ) = 0 such that v · · · ˆ v i · · · v n is an allowed elementary ( n − -path on G ;(ii). there exists at most one u ∈ V ( G ) with f ( u ) = 0 such that for some − ≤ j ≤ n , v · · · v j uv j +1 · · · v n (Specially, j = − , v · · · v j uv j +1 · · · v n = uv · · · v n ; j = n , v · · · v j uv j +1 · · · v n = v · · · v n u ) is an allowed elementary ( n + 1) -path on G . 7n an allowed elementary path v v · · · v n , the vertex v i is removable if i = 0 , or i = n ,or ≤ i ≤ n − and v i − → v i +1 is a directed edge in G . For example, in the digraph { v → v , v → v , v → v , v → v } , v is removable in the path v v v , while v is notremovable in the path v v v or the path v v v . Definition 4 (i) can be restated as follows:(i). in an allowed elementary path, at most one removable vertex has zero value.For the allowed elementary path v v · · · v n , a vertex u of G is called addable , if u → v isa directed edge, or v n → u is a directed edge, or there exists ≤ i ≤ n such that both v i − → u and u → v i +1 are directed edges. Definition 4 (ii) can be restated as follows:(ii). for an allowed elementary path, at most one addable vertex of G has zerovalue.Let f be a non-negative function on V ( G ) . For any allowed elementary path v v · · · v n ,we define the value of f on the path by letting f ( v v · · · v n ) = n X i =0 f ( v i ) . (3.1)We use γ ( n ) (or γ for short) to denote the allowed elementary n -path. For ≤ i ≤ n ,if v · · · ˆ v i · · · v n is an allowed elementary ( n − -path, then we write this ( n − -path as d i γ . Note that ˆ v · · · v i · · · v n and v · · · v i · · · ˆ v n are always allowed elementary ( n − -paths.Hence we always have d γ and d n γ . By (3.1), Definition 4 (i) can be restated as follows:(i). there exists at most one allowed elementary ( n − -path β ( n − such that f ( β ) = f ( γ ) and d i γ = β for some ≤ i ≤ n ;and Definition 4 (ii) can be restated as follows:(ii). there exists at most one allowed elementary ( n + 1) -path α ( n +1) such that f ( α ) = f ( γ ) and d i α = γ for some ≤ i ≤ n + 1 .For any allowed elementary paths γ and γ ′ , if γ ′ can be obtained from γ by removingsome vertices, then we write γ ′ < γ or γ > γ ′ . For any allowed elementary n -path γ ( n ) on G , we can rewrite (i), (ii) in Definition 4 equivalently as the following inequalities(i). n β ( n − < γ ( n ) | f ( β ) = f ( γ ) o ≤ ;(ii). n α ( n +1) > γ ( n ) | f ( α ) = f ( γ ) o ≤ .For an allowed elementary path γ , if in both (i) and (ii), the inequalities hold strictly, then γ is called critical. Precisely, Definition 5.
An allowed elementary n -path γ ( n ) is called critical , if both of the followingshold: (i)’ n β ( n − < γ ( n ) | f ( β ) = f ( γ ) o = 0 ,8ii)’ n α ( n +1) > γ ( n ) | f ( α ) = f ( γ ) o = 0 .By Definition 5, an allowed elementary n -path γ ( n ) on G is non-critical if and only ifeither of the followings holds:(i)” there exists β ( n − < γ ( n ) such that f ( β ) = f ( γ ) ;(ii)” there exists α ( n +1) > γ ( n ) such that f ( α ) = f ( γ ) . In this section, we prove some auxiliary results and some additional properties about discreteMorse functions and the critical allowed elementary paths of G . A directed loop in G is an allowed elementary path v v . . . v n v for n ≥ [1] . Lemma 4.1.
For any digraph G and any discrete Morse function f on G , if v v . . . v n v ( n ≥ ) is a directed loop in G , then f ( v i ) > strictly for any ≤ i ≤ n .Proof. Let α = v v . . . v n v be a directed loop in G . Suppose to the contrary, f ( v i ) = 0 for some ≤ i ≤ n . Suppose β = v i v i +1 · · · v n v · · · v i − v i , γ = v i v i +1 · · · v n v · · · v i − and γ ′ = v i +1 · · · v n v · · · v i − v i . Then γ < β, γ ′ < β and f ( β ) = f ( γ ) = f ( γ ′ ) . This contradictsthat f is a discrete Morse function on G . Therefore, f ( v i ) > strictly for any ≤ i ≤ n .The next lemma follows from Lemma 4.1. Lemma 4.2.
Let G be a digraph and f a discrete Morse function on G . Then any directedloop in G is critical.Proof. Let v v . . . v n v be an arbitrary directed loop in G . Suppose to the contrary, thedirected loop is non-critical. Then by Lemma 4.1, (i)” does not hold for the directed loop,hence (ii)” must hold for the directed loop. That is, there exists a vertex u of G such that f ( u ) = 0 and for some ≤ i ≤ n , v v . . . v i uv i +1 . . . v n v is a directed loop in G (here weuse the notation v n +1 = v ). This contradicts Lemma 4.1. Therefore, the directed loop v v . . . v n v is critical. Lemma 4.3.
Let G be a digraph and f a discrete Morse function on G . Then for anyallowed elementary path in G , there exists at most one index such that the correspondingvertex is with zero value. [1]. In [12, Definition 4.3], a loop on a digraph G is defined to be a based map from a line digraph to G such that the start vertex and the end vertex and the same. Our directed loop here is different from theloop in [12]. A directed loop is a special loop and the converse is not true. roof. Let α = v · · · v n be an allowed elementary path. Suppose to the contrary, f ( v i ) = f ( v j ) = 0 , i < j . There are two cases. Case 1 . v i = v j .Then we have β = v i · · · v j , γ = v i +1 · · · v j and γ ′ = v i · · · v j − such that β > γ , β > γ ′ and f ( β ) = f ( γ ) = f ( γ ′ ) . This contradicts that f is a discrete Morse function on G . Case 2 . v i = v j . Subcase 2.1 . j = i + 1 .Then we have f ( v i v j ) = f ( v i ) = f ( v j ) = 0 . This contradicts that f is a discrete Morsefunction on G . Subcase 2.2 . j ≥ i + 2 .Then v i v i +1 · · · v j is a directed loop with f ( v i ) = 0 . By Lemma 4.1, this is impossible.Therefore, the lemma follows. Lemma 4.4.
Let f be a discrete Morse function on digraph G . Then for any allowedelementary path in G , (i)” and (ii)” cannot both be true.Proof. Let γ = v v · · · v n be an allowed elementary path in G . Suppose to the contrary,by Definition 4, there must exist an allowed elementary ( n − -path β and an allowedelementary ( n + 1) -path α such that β < γ < α , f ( β ) = f ( γ ) and f ( α ) = f ( γ ) . ByProposition 2.6, we consider the following cases. Case 1 . There exists an allowed elementary n -path ˜ γ = γ such that β < ˜ γ < α .Then similar to the proof of [13, Lemma 2.5], by Definition 4, we have f ( β ) < f (˜ γ ) , f (˜ γ ) < f ( α ) . Thus f ( γ ) = f ( β ) < f (˜ γ ) < f ( α ) = f ( γ ) which is a contradiction. Case 2 . There does not exist any allowed elementary n -path ˜ γ = γ such that β < ˜ γ < α .Then β must be obtained by removing two subsequent vertices v i → v i +1 in α where ≤ i ≤ n . Hence, f ( v i ) = f ( v i +1 ) in α . This contradicts with Lemma 4.3.Therefore, (i)” and (ii)” cannot both be true.We call an allowed elementary path v v . . . v n simplicial if all the vertices v , v , . . . , v n are distinct. For each n ≥ , let S n ( G ) be the collection of all the formal linear combinationsof simplicial allowed elementary n -paths in G . Then S n ( G ) is a sub- R -module of P n ( G ) .The concatenation of a p -path α = v v · · · v p and a q -path β = w w · · · w q is α ∗ β = v v · · · v p w · · · w q , if v p = w , , if v p = w . Lemma 4.5. (a). Any allowed elementary path α in G is concatenations of simplicial al-lowed elementary paths β i and directed loops γ i : α = β ∗ γ ∗ β ∗ γ ∗ . . . ∗ β k − ∗ γ k − ∗ β k . (4.1)10 ere we allow each β i to be a single vertex in which case the corresponding concatena-tions would be trivial. Moreover, (4.1) is unique under a certain algorithm.(b). Let α be non-critical. Then in (4.1), there exists one β i which is non-critical.Proof. Firstly, we write α as a sequence of vertices v v . . . v n . If for each v i and each ≤ j ≤ i − , v i is different from v j , then α is simplicial. Thus we can let α = β .Otherwise, we let v i to be the first vertex such that there exists ≤ j ≤ i − satisfying v i = v j . Let β = v v . . . v j and γ = v j v j +1 . . . v i . Then v v . . . v i = β ∗ γ . Applythe same argument to v i v i +1 . . . v n . Since n is finite, by induction, α can be uniquelywritten as concatenations β ∗ γ ∗ β ∗ γ ∗ . . . ∗ β k − ∗ γ k − ∗ β k for some k under thisalgorithm. Hence (a) follows.Secondly, if α be non-critical, we consider two cases. Case 1 . (i)” holds for α .Then by Lemma 4.4, (ii)” does not hold for α . By Lemma 4.2 and Lemma 4.3, thereexists unique vertex with zero value which belongs to some β i in (4.1) and does not belongto any directed loop. Hence β i is non-critical. Case 2 . (i)” does not hold for α .Then by Lemma 4.4, (ii)” holds for α . By Lemma 4.2 and Lemma 4.3, there exists uniquevertex u ∈ V ( G ) with zero value such that u is addable in α , but not addable in any directedloop. Thus u is addable in some β i , which makes β i non-critical.Combining above cases, (b) follows.By Lemma 4.5, it is direct to see that any digraph must satisfy one of the followingconditions:(A). For each allowed elementary path α , all β i in (4.1) are simplicial allowed elementarypaths satisfying that each vertex of β i belongs to one of the directed loops in (4.1);(B). For an allowed elementary path α ′ , there exists a β ′ i in (4.1) satisfying that at leastone vertex of β i does not belong to any directed loop in (4.1).Then we claim that Proposition 4.6.
For digraphs satisfying the condition (A), all discrete Morse functionsare trivial; For digraphs satisfying the condition (B), there are not only trivial functions, butalso nontrivial functions.Proof.
Let G be a digraph. If G satisfies the condition (A), then by Lemma 4.1, each vertexof G can only be assigned a positive real number. Hence, all discrete Morse functions on G are trivial here.If G satisfies the condition (B), we can define trivial functions on G firstly. In addition,we can assign to one vertex which does not belong to any directed loop in (4.1), and assignpositive real values to other vertices on G . By Definition 4, we know that the function11efined in this way is a discrete Morse function on G . That is, for this case, there are notonly trivial functions on G , but also nontrivial functions. Example 4.7.
Let α = v v v v v v v v v be an allowed elementary path of G . Then itcan be written as α = β ∗ γ ∗ β where β = v v v v , γ = v v v and β = v v v v ,or α = β ′ ∗ γ ′ ∗ β ′ where β ′ = v v v v v , γ ′ = v v v v and β ′ = v v . But under thealgorithm of Lemma 4.5(a), it can be written as β ∗ γ ∗ β uniquely. Let G ′ be a sub-digraph of G . • Suppose there is a discrete Morse function f on G . Then by Definition 4, by defining f ′ ( v ) = f ( v ) for any v ∈ V ( G ′ ) , f gives a discrete Morse function f ′ on G ′ . Hence The restriction of a discrete Morse function to a sub-digraph is still a discreteMorse function. • Suppose there is a discrete Morse function f ′ on G ′ . The next example shows that f ′ may not be extendable to be a discrete Morse function on G . Example 4.8.
Let V = { v , v , v , v } . Let G ′ be a digraph with the set of vertices V andthe set of directed edges { v → v , v → v , v → v } . Let f ′ be a function on V given by f ′ ( v ) = f ′ ( v ) = 0 and f ′ ( v ) = 1 , f ′ ( v ) = 2 . Then f ′ is a discrete Morse function on G ′ .Let G = { V ( G ) , E ( G ) ∪ { v → v }} .Then G ′ is a sub-digraph of G . However, f ′ is not adiscrete Morse function on G . Since V ( G ) = V ( G ′ ) , there does not exist any discrete Morsefunction f on G such that the restriction of f to G ′ equals f ′ . For each n ≥ , let Crit n ( G ) be the free R -module consisting of all the formal linearcombinations of critical allowed elementary n -paths on G . Then Crit n ( G ) is a sub- R -moduleof P n ( G ) . Proposition 4.9.
Let G , G ′ both be digraphs such that G ′ ⊆ G . Let f be a discrete Morsefunction on G and f ′ = f | G ′ . Then for each n ≥ , Crit n ( G ) ∩ P ( G ′ ) ⊆ Crit n ( G ′ ) .Proof. Let α = v · · · v n be a critical allowed elementary path in G . Assume α is also anallowed elementary path in G ′ . Then for any ≤ i ≤ n , if d i ( α ) is allowed in G ′ , we havethat f ′ ( d i ( α )) = f ( d i ( α )) < f ( α ) = f ′ ( α ) by Definition 5(i)”. Moreover, for any u ∈ V ( G ′ ) ,if v · · · v j uv j +1 · · · v n is an allowed elementary path in G ′ , by Definition 5(ii)”, we have that f ′ ( v · · · v j uv j +1 · · · v n ) = f ( v · · · v j uv j +1 · · · v n ) > f ( α ) = f ′ ( α ) . This implies the assertion.The next example shows that the inverse of Proposition 4.9 may not be true.12 xample 4.10.
Let V = { v , v , v } . Let G be a digraph with the set of vertices V andthe set of directed edges { v → v , v → v , v → v } . Let f be a function on V givenby f ( v ) = 0 and f ( v ) = 1 , f ( v ) = 2 . Then f is a discrete Morse function on G and α = v v v is not critical in G . Let G ′ be the digraph with the set of vertices V and the setof directed edges { v → v , v → v } . Then G ′ is a sub-digraph of G and f ′ = f | G ′ is adiscrete Morse function on G ′ . However, α = v v v is critical in G ′ . In this section, we define the discrete gradient vector fields on digraphs and prove that it isan acyclic matching. Based on this, we give the proof of Theorem 1.1.
Let G be a digraph. Let f : V ( G ) −→ [0 , + ∞ ) be a discrete Morse function on G . Forany n ≥ and any allowed elementary paths α ( n ) < β ( n +1) on G , if f ( α ) = f ( β ) , then weassign a pair ( α, β ) . By collecting all such pairs, we obtain a partial matching M ( G, f ) .We call M ( G, f ) the (combinatorial) discrete gradient vector field of f . The properties ofa discrete Morse function imply that each allowed elementary path of G is in at most onepair of M ( G, f ) .For each n ≥ , by the (combinatorial) discrete gradient vector field M ( G, f ) , we canconstruct the (algebraic) discrete gradient vector field grad f which is an R -linear map from P n ( G ) to P n +1 ( G ) . For an allowed elementary n -path α ( n ) on G , if there exists an allowedelementary ( n + 1) -path β ( n +1) such that ( α, β ) ∈ M ( G, f ) , then we set ( grad f )( α ) = −h ∂β, α i β. If there does not exist such β , then we set ( grad f )( α ) = 0 . We extend grad f linearly over R and obtain an R -linear map grad f : P n ( G ) −→ P n +1 ( G ) .We call grad f the (algebraic) discrete gradient vector field of f . By Lemma 4.4 and a similarargument with [13, Theorem 6.3 (1)], it followsgrad f ◦ grad f = 0 . Let R be an arbitrary commutative ring with a unit. Let G be a digraph. Consider thechain complex · · · ∂ n +2 −→ Ω n +1 ( G ) ∂ n +1 −→ Ω n ( G ) ∂ n −→ · · · n ≥ , choose a basis B n ( G ) for Ω n ( G ) . For b ∈ B n ( G ) and a ∈ B n − ( G ) , define h ∂ n b, a i to be the coefficients of the term a in the linear combination ∂ n b . Then ∂ n b = X a ∈ B n − ( G ) h ∂ n b, a i a. Definition 6. (cf. [17, Definition 1.1]) A partial matching
M ⊆ S n ≥ B n − ( G ) × B n ( G ) is a collection of pairs ( a, b ) such that a ∈ B n − ( G ) , b ∈ B n ( G ) for some n , and h ∂ n b, a i isinvertible in R . Write m ( b > a ) = h ∂ n b, a i for ( a, b ) ∈ M .For each n ≥ , consider the subsets of B n ( G ) U ( B n ( G )) = { b ∈ B n ( G ) | there exists a ∈ B n − ( G ) such that ( a, b ) ∈ M} , D ( B n ( G )) = { a ∈ B n ( G ) | there exists b ∈ B n +1 ( G ) such that ( a, b ) ∈ M} , C ( B n ( G )) = B n ( G ) \ (cid:0) U ( B n ( G )) [ D ( B n ( G )) (cid:1) . Given b ∈ B n ( G ) and a ∈ B n − ( G ) , an alternating path is a sequence b > a < b > a < b > a · · · > a k < b k > a such that for each i = 1 , , · · · , k , ( a i , b i ) ∈ M . For an alternating path p , we write p • = b , p • = a and define m ( p ) = ( − k m ( b > a ) m ( b > a ) · · · m ( b k > a ) m ( b > a ) m ( b > a ) · · · m ( b k > a k ) . Definition 7. (cf. [17, Definition 1.2]) A partial matching M is called acyclic, if there doesnot exist any cycle a < b > a < b > a < · · · > a k < b k > a with k ≥ and all b i ∈ S ( B n ( G )) (for any n ≥ ) are distinct.Let M be an acyclic partial matching. Definition 8. (cf. [17, Definition 1.4]) The Morse complex is defined as · · · ∂ M n +2 −→ C M n +1 ( B ∗ ( G )) ∂ M n +1 −→ C M n ( B ∗ ( G )) ∂ M n −→ C M n − ( B ∗ ( G )) ∂ M n − −→ · · · , where the R -module C M n ( B ∗ ( G )) is freely generated by the elements of C ( B n ( G )) , and theboundary map is given by ∂ M n ( b ) = P p m ( p ) p • for all alternating paths p with p • = b .An atom chain complex is a chain complex · · · −→ −→ R id −→ R −→ −→ · · · wherethe only nontrivial modules are in the dimensions d and d − , and the boundary map is theidentity map. Such an atom chain complex is denoted by Atom ( d ) (cf. [17, P.870]). Lemma 5.1. (cf. [17, Theorem 2.1], [18, Theorem 2.2]) Assume that we have a free chaincomplex with a basis (Ω ∗ , B ∗ ) , and an acyclic matching M . Then(a). Ω ∗ decomposes as a direct sum of chain complexes C M∗ ( B ∗ ( G )) L T ∗ , where T ∗ ≃ L ( a,b ) ∈M Atom ( dim b ) ; b). H ∗ (Ω ∗ ) = H ∗ ( C M∗ ( B ∗ ( G ))) . To prove Theorem 1.1, we first prove the next lemma to show that M ( G, f ) is acyclic. Lemma 5.2.
Let G be a digraph and f a discrete Morse function on G . Then M ( G, f ) isan acyclic matching.Proof. By considering the value of f at each vertex of G , we separate the proof into twocases. Case 1 . f ( v ) > for any vertex v ∈ V ( G ) .Then we have M ( G, f ) = ∅ . Case 2 . There exists a vertex v ∈ V ( G ) such that f ( v ) = 0 .Then M ( G, f ) is a nonempty finite set. That is, for each n ≥ , there exist finite pairs { α ( n ) , β ( n +1) } such that α ( n ) < β ( n +1) and f ( α ( n ) ) = f ( β ( n +1) ) .Suppose to contrary, by Definition 7, there exists a cycle α ( n )1 < β ( n +1)1 > α ( n )2 < β ( n +1)2 > α ( n )3 < · · · > α ( n ) k < β ( n +1) k > α ( n )1 with k ≥ , all β ( n +1) i are distinct and ( α ( n ) i < β ( n +1) i ) ∈ M ( G, f ) ( ≤ i ≤ k ) for aninteger n ≥ . Since ( α ( n ) i , β ( n +1) i ) ∈ M ( G, f ) , β ( n +1) i is non-critical for each ≤ i ≤ k . ByLemma 4.4, (ii)” does not hold for β ( n +1) i . That is, there exists one vertex u i of β ( n +1) i suchthat f ( u i ) = 0 for each ≤ i ≤ k . We consider the following two subcases. Subcase 1 . α ( n )2 = α ( n )1 .Then by Definition 4(ii), we have that β ( n +1)1 = β ( n +1)2 . This contradicts with that all β ( n +1) i are distinct. Subcase 2 . α ( n )2 = α ( n )1 .Then V ( α ( n )2 ) = V ( β ( n +1)1 ) \{ a nonzero vertex of β ( n +1)1 } and u ∈ V ( α ( n )2 ) . By Lemma 4.3,since α ( n )2 < β ( n +1)2 , it follows that f ( α ( n )2 ) < f ( β ( n +1)2 ) . This contradicts with ( α ( n )2 , β ( n +1)2 ) ∈M ( G, f ) .Summarising Case 1 and Case 2, we have that M ( G, f ) is acyclic.Now we prove Theorem 1.1. Proof of Theorem 1.1.
Firstly, by Proposition 4.6, we have that on any digraph there alwaysexists a discrete Morse function. Secondly, since G is transitive, by Proposition 2.1, we have Ω n ( G ) = P n ( G ) for each n ≥ . Hence, all allowed elementary paths form a basis of thechain complex { Ω ∗ ( G ) , ∂ ∗ } . Therefore, by Lemma 5.2, by taking the allowed elementarypaths as a basis of Ω ∗ ( G ) , there always exists an acyclic matching. Hence by Lemma 5.1(a),(1.1) follows.Moreover, by Lemma 5.1(b), the homology groups of Ω ∗ ( G ) and C M∗ ( B ∗ ( G )) are iso-morphic. Thus (1.2) follows. 15n addition, we can get the following theorem. Theorem 5.3.
Let G be a digraph containing neither triangle nor square. Then both (1.1)and (1.2) hold.Proof. By [7, Theorem 4.3], dim Ω n ( G ) = 0 for all n ≥ . Meanwhile, Ω ( G ) = P ( G ) and Ω ( G ) = P ( G ) . Hence, by Lemma 5.2 and [17, Theorem 2.1], the assertion follows. Remark : Let G be a digraph (not transitive) containing triangles or squares. Foreach n ≥ , the elements of the basis of Ω n ( G ) are not only allowed elementary paths,but also linear combinations of allowed elementary paths. The existence of acyclicmatching needs further explorations. M -collapses by discrete gradient vector fields Let G be a digraph and f a discrete Morse function on G . For any vertex v ∈ V ( G ) , wedefine the number of edges starting from v as the out-degree of v , and the number of edgesarriving at v as the in-degree of v . Assume that the out-degree and in-degree of any zero-point of f on G are both 1. In this section, we define M -collapses and give the proof ofTheorem 1.2.Let v ∈ G be a zero-point of f on G . By Lemma 4.2, v is not the vertex of any directedloop. Moreover, since the out-degree and in-degree of any zero-point of f on G are both 1,there exists a unique ordered triple of vertices ( u, v, w ) such that u → v → w and u, v, w are distinct. If u → w is a directed edge in G , then we substitute u → v → w with u → w .Consider the digraph G ′ whose set of vertices is V ( G ) \ { v } and whose set of directed edgesis E ( G ) \{ u → v, v → w } . By Definition 4, the function f ′ on V ( G ′ ) defined by f ′ ( x ) = f ( x ) for any x ∈ V ( G ′ ) gives a discrete Morse function on G ′ . We call the pair ( G ′ , f ′ ) a one-step M -collapse of the pair ( G, f ) . Lemma 6.1.
For any pair ( α, β ) ∈ M ( G, f ) , it is in one of the following forms: ( α, β ) = α = · · · → u → w → · · · , β = · · · → u → v → w → · · · α = · · · → u, β = · · · → u → vα = w → · · · , β = v → w → · · · where f ( v ) = 0 and u → v → w .Proof. Since ( α, β ) ∈ M ( G, f ) , β ( n +1) > α ( n ) and f ( α ) = f ( β ) . By Lemma 4.3, there mustexist a unique vertex v of β such that f ( v ) = 0 . Since the out-degree and in-degree of anyzero-point of f on G are both 1, there exists a unique ordered triple of vertices ( u, v, w ) suchthat u → v → w and u, v, w are distinct. There are two cases to consider. Case 1 . u → w is a directed edge in G . Subcase 1.1 . u → v → w is a subgraph of β .16hen α is the allowed elementary path obtained by substituting u → v → w with u → w . Subcase 1.2 . u → v → w is not a subgraph of β .Then we must have β = · · · → u → v or β = v → w → · · · . Correspondingly, α = · · · → u or α = w → · · · . Case 2 . u → w is not a directed edge in G . Subcase 2.1 . u → v → w is a subgraph of β .Then by Lemma 4.3 and Definition 5, α is critical. This contradicts that ( α, β ) ∈M ( G, f ) . Subcase 2.2 . u → v → w is not a subgraph of β .Then we must have β = · · · → u → v or β = v → w → · · · . It follows that α = · · · → u and α = w → · · · respectively.Therefore, the lemma follows. Lemma 6.2. M ( G ′ , f ′ ) ⊆ M ( G, f ) .Proof. For any pair ( α ′ , β ′ ) ∈ M ( G ′ , f ′ ) , α ′ , β ′ are both allowed elementary paths in G ′ such that α ′ < β ′ and f ′ ( α ′ ) = f ′ ( β ′ ) . Since G ′ ⊆ G and f ′ = f | G ′ , it follows that α ′ , β ′ are also allowed elementary paths in G and f ( α ′ ) = f ( β ′ ) . Hence ( α ′ , β ′ ) ∈ M ( G, f ) . Thelemma follows.Since there are finite zero-points of f on G , it follows that there are finite triples { ( u k , v k , w k ) } ≤ k ≤ N such that f ( v k ) = 0 and u k → w k is an allowed elementary pathof G . Denote the subgraph obtained by k -step M -collapse as G k whose set of vertices is V ( G ) \{ v , · · · , v k } and whose set of directed edges is E ( G ) \{ ( u → v , v → w ) , · · · , ( u k → v k , v k → w k ) } . Similarly, the restriction of f on G k (denoted as f k ) is a discrete Morse func-tion on G k . Obviously, G N ⊆ · · · ⊆ · · · G ⊆ G . By induction, we can get a subgraph of G in which there is no triple ( u, v, w ) such that f ( v ) = 0 , u → v → w and u → w . We denoteit as ( ˜ G, ˜ f ) .By Lemma 6.1 and Lemma 6.2, we have that Proposition 6.3.
Any pair ( α, β ) ∈ M ( ˜ G, ˜ f ) is in the form of α = · · · → uβ = · · · → u → v or α = w → · · · β = v → w → · · · where f ( v ) = 0 , u → v → w and u → w is not a directed edge in G .Proof. By Lemma 6.2, M ( ˜ G, ˜ f ) ⊆ M ( G, f ) . Note that any pair ( α, β ) ∈ M ( G, f ) in Case1of Lemma 6.1 is removed by M -collapse. Hence the assertion follows.In the next, we prove that ˜ G and G have the same path homology groups.17 roof of Theorem 1.2. Define a digraph map r : G → ˜ G such that r ( v ) = w, if f ( v ) = 0 and there exists a triple ( u, v, w ) such that u → v → w and u → w ; v, otherwise . (6.1)By the definition of M -collapses, it can be verified directly that r | ˜ G = id ˜ G and r is aretraction of G onto ˜ G . By Proposition 2.2 and Theorem 2.3, it is sufficient to prove that i ◦ r ≃ id G , where i : ˜ G → G is the natural inclusion map.Let I be the line digraph such that the set of vertices is { , } and the set of directededges is exactly { → } . Define a map F : G ⊡ I → G such that F ( v,
0) = v and F ( v,
1) = ( i ◦ r )( v ) . Then by (6.1), we have that F | G ⊡ { } = id G , F | G ⊡ { } = i ◦ r . Without loss of generality, we suppose that F is a one-step M -collapse.Then there exist a vertex v such that f ( v ) = 0 and a unique ordered triple of vertices ( u, v, w ) with u → v → w and u → w in G . It follows that F (( u, → ( v, u → v, F (( v, → ( w, v → w, F (( u, → ( w, u → w,F (( u, → ( v, u → w, F (( v, → ( w, w, F (( u, → ( w, u → w,F (( u, → ( u, u, F (( v, → ( v, v → w, F (( w, → ( w, w. Hence F is well-defined and it is a digraph map from G ⊡ I to G . By Definition 2, wehave that i ◦ r ≃ id G . By Definition 3, r is a deformation retraction. This implies thetheorem. Acknowledgement . The authors would like to thank Prof. Yong Lin and Prof. Jie Wufor their supports, discussions and encouragements. The authors also would like to expresstheir deep gratitude to the reviewer(s) for their careful reading, valuable comments, andhelpful suggestions.The first author is supported by the Youth Fund of Hebei Provincial Department ofEducation (QN2019333), the Natural Fund of Cangzhou Science and Technology Bureau(No.197000002) and a Project of Cangzhou Normal University (No.xnjjl1902). The secondauthor is supported by the Postdoctoral International Exchange Program of China 2019project from The Office of China Postdoctoral Council, China Postdoctoral Science Foun-dation.
References [1] R. Ayala, L.M. Fern ´ a ndez and J.A. Vilches, Discrete Morse inequalities on infinitegraphs . Electron. J. Combin. (1) (2009), R38.182] R. Ayala, L.M. Fern ´ a ndez and J.A. Vilches, Morse inequalities on certain infinite 2-complexes . Glasg. Math. J. (2) (2007), 155-165.[3] R. Ayala, L.M. Fern ´ a ndez, D. Fern ´ a ndez-Ternero and J.A. Vilches. Discrete Morsetheory on graphs . Topol. Appl. (2009), 3091-3100.[4] R. Ayala, L.M. Fern ´ a ndez, A. Quintero and J.A. Vilches, A note on the pure Morsecomplex of a graph . Topol. Appl. (2008), 2084-2089.[5] J. Bang-Jensen and G.Z. Gutin,
Digraphs: Theory, Algorithms and Applications . 2-ndEdition. Springer Monographs in Mathematics, Springer, 2009.[6] S. Bressan, J. Li, S. Ren and J. Wu,
The embedded homology of hypergraphs and appli-cations . Asian J. Math. (3) (2019), 479-500.[7] A. Grigor’yan, Y. Lin, Y. Muranov and S.T. Yau, Homologies of path complexes anddigraphs . Preprint arXiv: 1207. 2834v4 (2013).[8] A. Grigor’yan, Y. Lin, Y. Muranov and S.T. Yau,
Cohomology of digraphs and (undi-rected) graphs . Asian J. Math. (5) (2015), 887-932.[9] A. Grigor’yan, Y. Lin, Y. Muranov and S.T. Yau, Path complexes and their homologies path homology theory of multi-graphs and quivers . Forum Math. (5) (2018), 1319-1337.[11] A. Grigor’yan, Y. Muranov and S.T. Yau, Homologies of digraphs and K ¨ u nneth formu-las . Commun. Anal. Geom. (5) (2017), 969-1018.[12] A. Grigor’yan, Y. Lin, Y. Muranov and S.T. Yau, Homotopy theory for digraphs . PureAppl. Math. Q. (4) (2014), 619-674.[13] R. Forman, Morse theory for cell complexes . Adv. Math. (1998), 90-145.[14] R. Forman,
Discrete Morse theory and the cohomology ring . Trans. Amer. Math. Soc. (12) (2002), 5063-5085.[15] R. Forman,
A user’s guide to discrete Morse theory . Sém. Lothar. Combin (2002),35pp.[16] R. Forman, Witten-Morse theory for cell complexes . Topology (5) (1998), 945-979.[17] Dmitry N. Kozlov, Discrete Morse theory for free chain complexes . C. R. Acad. Sci.Paris, Ser. I (2005) 867-872.[18] M. J ¨ o llenbeck and V. Welker, Minimal resolutions via algebraic discrete Morse theory .Memoirs of the American Mathematical Society , 2009.1919] H. Kannan, E. Saucan, I. Roy and A. Samal,
Persistent homology of unweighted complexnetworks via discrete Morse theory . Scientific Reports (2019), article 13817.[20] T. Lewiner, H. Lopes and G. Tavares, Applications of Forman’s discrete Morse the-ory to topology visualization and mesh compression . Transactions on visualization andcomputer graphics (5) (2004), 499-508, IEEE.[21] K. Mischaikow and V. Nanda, Morse theory for filtrations and efficient computation ofpersistent homology . Discrete Comput. Geom. (2013), 330-353.Chong Wang (for correspondence)Address: School of Mathematics, Renmin University of China, 100872 China.2