Graphs are widely used abstractions to model systems and interactions, e.g., in molecular chemistry [1], biochemistry [2], systems biology [3] [4], neuroscience [5], and social sciences [6]. Therefore, graph theoretical problems play a pivotal role across all disciplines of sciences. However, the combinatorical complexity of graph theoretical problems often hampers practical applications. Despite trees, planar graphs are a graph class for which most optimization or decision problems can efficiently be solved or closely be approximated. This is notably the case for the graph isomorphism problem [7], graph-coloring problems [8] and detection of Hamilton cycles [9], i.e., solving the traveling salesman problem [10]. These facts are reflected in Whitney’s theorem [11] that proves planar embeddings of 3-connected graphs to be unique up to isomorphism. However, planarity is a very strong condition on the graph G, which often does not match to instances occurring in practice. Extending considerations to graphs of higher genus allow to relax the conditions on the graph structure, which, by itself, is generated by the cycles the graph G contains. However, as counting all elementary cycles is already a #P-hard [12] problem finding an embedding of minimum genus is NP-hard [13]. In addition, the number of possible embeddings increases exponentially with the graph genus [13], which in contrast to Whitney’s Theorem makes it hard to efficiently represent G independently of its automorphism class.

Here, we address the question of whether efficiently computable invariants of graphs exist and how they can be used to measure the structural complexity of graphs independently of their representation. Biggs Theorem [14] is a classic result in graph theory that, similarly to the Euler characteristic of surfaces [15], provides such an invariant for simple digraphs.

Theorem (Biggs Theorem). The set of connected cycles O ¯ ( G ) of a simple digraph G generates a free R-module or vector space Λ ( G , R ) of dimension

dim R Λ ( G , R ) = | E | − | V | + # G   ,   R = ℤ , ℤ 2   , (1)

where #G denotes the connected components of G.

However, Biggs Theorem is only known to hold for the space generated by connected cycles of a simple digraph. Here we generalize the statement to the case of the directed cycle space 〈 O ( G ) 〉 R , R = ℤ , ℤ 2 spanned by all directed cycles O ( G ) of a multi-digraph G.

To provide an algebraic notion of cycles we define the characteristic vector representing how often and in which direction an edge is passed by a cycle or path.

Definition 2 (Characteristic Vector). Let G = ( V , E , head , tail ) be a multi-digraph we introduce the free R-modules, R = ℤ , ℤ 2

V ( G ) : = ⊕ v ∈ V   ℤ v ,   V 2 ( G ) : = V ( G ) / 2 V ( G ) = ⊕ v ∈ V   ℤ 2 v (4)

E ( G ) : = ⊕ e ∈ E   ℤ e   ,   E 2 ( G ) : = E ( G ) / 2 E ( G ) = ⊕ e ∈ E   ℤ 2 e   . (5)

Let ε = { e 1 , ⋯ , e n } , e i ∈ E , i = 1, ⋯ , n ∈ ℕ be an ordered list of connected edges. Then for i = 1 , ⋯ , n − 1 we define

x i = { 1   , if     e i , e i + 1     are consecutive − 1   , if     e i , e i + 1     are connected but not consecutive (6)

If ε ∈ O ¯ ( G ) is a cycle then we set x n = 1 if e n , e 1 are consecutive and x n = − 1 else. If ε ∈ O ¯ ( G ) then we set x n = 0 . By considering y i = x i mod 2 ℤ , i = 1 , ⋯ , n the vectors

[ ε ] = ∑ i = 1 , ⋯ , n     x i e i ∈ E ( G ) ,   [ ε ] 2 = ∑ i = 1 , ⋯ , n     y i e i ∈ E 2 ( G ) (7)

are called the characteristic vectors of ε with respect to ℤ -and ℤ 2 -coefficients.

Remark 1. We want the following fact to be clear. The characteristic vector depends on the ordering of ε . However, if c 1 , c 2 ∈ O ¯ el ( G ) are connected elementary cycles such that c 1 ∼ c 2 are equivalent. Then [ c 1 ] = ± [ c 2 ] holds. If c 1 , c 2 ∈ O el ( G ) are directed then even [ c 1 ] = [ c 2 ] holds. Thus, the maps

χ : O el ( G ) / ∼ → E ( G )   ,       c ↦ [ c ]   ,       χ 2 : O ¯ el ( G ) / ∼   → E 2 ( G )   ,       c ↦ [ c ] 2 (8)

are well defined.

Definition 3 (Incidence operator). Let G = ( V , E , head , tail ) be a multi-digraph. We denote with I ( G , ℤ ) : E → V , I ( G , ℤ 2 ) : E 2 → V 2 the ℤ , ℤ 2 -linear incidence operator operators, defined on the generators by

I ( G , ℤ ) [ e ] = head ( e ) − tail ( e ) , (9)

I ( G , ℤ 2 ) [ e ] 2 = head ( e ) + tail ( e ) mod 2 V ( G ) . (10)

By choosing an enumeration V ≅ { v 1 , ⋯ , v | V | } , E = { e 1 , ⋯ , e | E | } we can represent I ( G , ℤ ) by the classic incidence matrix I ( G , ℤ ) = ( θ i j ) 1 ≤ i ≤ | E | 1 ≤ j ≤ | V | with

θ i j = { 0, if     tail ( e i ) = v j       and     head ( e i ) = v j , 0, if     tail ( e i ) ≠ v j       and       head ( e i ) ≠ v j , 1, if     tail ( e i ) = v j       and       head ( e i ) ≠ v j , − 1, if     tail ( e i ) ≠ v j       and       head ( e i ) = v j . (11)

The notion of I ( G , ℤ 2 ) = ( μ i j ) 1 ≤ i ≤ | E | 1 ≤ j ≤ | V | is given by setting μ i j = θ i j mod 2 ℤ .

By Remark 1 the characteristic vector [ c ] , c ∈ O ¯ ( G ) might depend on the ordering of c. However, the kernel of I ( G , R ) is not sensitive to the possible choices of signs. Therefore, we obtain a well defined algebraic notion of cycles as follows.

Lemma 3. Let G = ( V , E , head , tail ) be a multi-digraph and O ⊆ O ¯ el ( G ) be a set of elementary cycles. Then we denote with

[ O ] = { x ∈ E ( G ) | x = [ c ] , c ∈ O }   , (12)

[ O ] 2 = { x ∈ E 2 ( G ) | x = [ c ] 2 , c ∈ O } (13)

the set of all characteristic vectors induced by cycles in O. The free generated cycle modules are given by

Λ ( G , ℤ ) : = 〈 [ O ¯ ( G ) ] 〉 ℤ = 〈 [ O ¯ el ( G ) ] 〉 ℤ = ker I ( G , ℤ )

Λ ( G , ℤ 2 ) : = 〈 [ O ¯ ( G ) ] 2 〉 ℤ 2 = 〈 [ O ¯ el ( G ) ] 2 〉 ℤ 2 = ker I ( G , ℤ 2 ) ,

where 〈 ⋅ 〉 ℤ , 〈 ⋅ 〉 ℤ 2 denotes the span with respect to ℤ , ℤ 2 .

Proof. The identities 〈 [ O ¯ ( G ) ] 〉 ℤ = 〈 [ O ¯ el ( G ) ] 〉 ℤ ,   〈 [ O ¯ ( G ) ] 2 〉 ℤ 2 = 〈 [ O ¯ el ( G ) ] 2 〉 ℤ 2 follow directly by the definition of [ O ¯ ( G ) ] , [ O ¯ ( G ) ] 2 . Further, we verify that for any list ε = { e 1 , ⋯ , e n } ⊆ E of connected edges, due to (6), we have

± I ( G , ℤ ) [ ε ] = tail ( e 1 ) − head ( e n )   ,

I ( G , ℤ 2 ) [ ε ] 2 = tail ( e 1 ) + head ( e n ) mod 2 V ( G ) . (14)

Hence,

I ( G , ℤ ) [ ε ] = 0       ⇔       ε ∈ O ¯ ( G )   ,   I ( G , ℤ 2 ) [ ε ] 2 = 0       ⇔       ε ∈ O ¯ ( G ) (15)

proving the claim.

To characterize the cycle spaces we rephrase Biggs Theorem [14] for connected cycles of simple digraphs matching our setup and notation.

Theorem 4 (Biggs Theorem). Let G = ( V , E , head , tail ) be a simple digraph then the dimension of the connected cycle spaces are given by

dim R Λ ( G , R ) = | E | − | V | + # G ,   R = ℤ , ℤ 2 , (16)

where #G denotes the number of connected components of G.

Indeed, for simple digraphs the first line of the incidence matrix in (11) is unnecessary and thereby our definition yields the classic notion of I ( G , R ) for any fixed enumeration of V , E . In fact, the proof of Theorem 4 is based on computing the rank of I ( G , R ) , which is independent of the chosen enumeration, yielding our reformulation to be genuine. Due to Lemma 3 we obtain the following consequence.

Corollary 1 (Biggs Theorem for connected cycles of multi-digraphs). Let G = ( V , E , head , tail ) be a multi-digraph. Then the dimension of the connected cycle spaces are given by

dim R Λ ( G , R ) = | E | − | V | + # G ,   R = ℤ , ℤ 2 , (17)

where #G denotes the number of connected components of G.

Proof. Indeed, each loop or multiple edge increases the dimension by 1. More precisely: Every loop [ { e } ] , [ { e } ] 2 or two cycle [ { e , f } ] , [ { e , f } ] 2 , e , f ∈ E is R-linear independent, to all other elements in [ O ¯ ( G \ e ) ] , [ O ¯ ( G \ e ) ] 2 . Thus, by removing e we obtain dim R Λ ( G \ e , R ) = dim R Λ ( G , R ) − 1 . Recursively applying this procedure ends up with a simple digraph G ′ = ( V , E ′ , hea d ′ , tai l ′ ) with # G ′ = # G . Thus, due to Theorem 4 we compute

dim R Λ ( G , R ) = | E | − | E ′ | + dim R Λ ( G ′ , R ) = | E | − | E ′ | + | E ′ | − | V | + # G ′ = | E | − | V | + # G   . (18)

How to generalize the characterisation of cycle spaces in the directed case is provided in the next section.

We prove a generalization, we introduce the concept of contracting edges in a multi-digraph as follows.

Definition 4 (Contracted Graph). Let G = ( V , E , head , tail ) be a multi-digraph and e ∈ E . Let v = head ( e ) , u = tail ( e ) . An equivalence relation ∼ u , v on V is defined by

x ∼ u , v y   ⇔   x = y       or       x , y ∈ { u , v } . (19)

The equivalence class of x ∈ V is denoted by [ x ] ∼ u , v and V / e : = V / ∼ u , v gives the quotient of V with respect to ∼ u , v . Further, we define E / e : = E \ { e } and set

head / e ( f ) = [ head ( f ) ] ∼ u , v   ,   tail / e ( f ) = [ tail ( f ) ] ∼ u , v   ,   ∀   f ∈ E / e   . (20)

The multi-digraph G / e : = ( V / e , E / e , head / e , tail / e ) is called the contracted graph of G with respect to e ∈ E . If e , f ∈ E then we observe that ( G / e ) / f ≅ ( G / f ) / e . Thus, if G ′ ⊆ G is a subgraph then G / G ′ : = G / E ( G ′ ) can be defined by contracting E ( G ′ ) in arbitrary order, yielding a well defined notion.

Remark 2. Let G = ( V , E , head , tail ) be a simple digraph and e , f , h ∈ E such that e , f are consecutive and tail ( h ) = tail ( e ) , head ( h ) = head ( f ) . Then G / e becomes a multi-digraph, i.e., h and f become parallel edges in G / e . Furthermore, the contraction of a two cycle c = { e , f } , tail ( e ) ∈ { tail ( f ) , head ( f ) } , head ( e ) ∈ { tail ( f ) , head ( f ) } results in a loop c / e ∈ O ¯ ( G / e ) . These phenomena are the reasons why we consider multi-digraphs with loops, rendering the notion of contracted graphs consistent within our framework. Certainly, contraction preserves connectivity, i.e., # G = # G / e .

Now we have all ingredients to state the first main theorem of this article.

Theorem 5 (Biggs Theorem for directed cycles). Let G = ( V , E , head , tail ) be a multi-digraph and Λ ( G , R ) = 〈 O ¯ el ( G ) 〉 R its cycle space with respect to R = ℤ , ℤ 2 . Then

G ( O el ( G ) ) = G ( O ¯ el ( G ) ) if and only if 〈 O el ( G ) 〉 R = 〈 O ¯ el ( G ) 〉 R . (21)

Consequently, in case of equivalence, there holds

dim R 〈 O el ( G ) 〉 R = | E | − | V | + # G = dim R Λ ( G , R )   . (22)

Proof. The proof splits into several steps:

Step 1: We show 〈 O el ( G ) 〉 ℤ = 〈 O ¯ el ( G ) 〉 ℤ ⇒ G ( O el ( G ) ) = G ( O ¯ el ( G ) ) . Indeed, given [ c ] = ∑ i = 1 n     x i e i ∈ 〈 O ¯ el ( G ) 〉 ℤ . Then we can generate [ c ] = ∑ k = 1 m     λ k [ c k ] by characteristic vectors of directed elementary cycles c k ∈ O el ( G ) , k = 1, ⋯ , m . Thus, every e i hast to be contained in at least one of them proving G ( O el ( G ) ) = G ( O ¯ el ( G ) ) .

Step 2: We show G ( O el ( G ) ) = G ( O ¯ el ( G ) ) ⇒ 〈 O el ( G ) 〉 ℤ = 〈 O ¯ el ( G ) 〉 ℤ . Indeed, O el ( G ) ⊆ O ¯ el ( G ) implies that by Lemma 3 〈 O el ( G ) 〉 ℤ ⊆ 〈 O ¯ el ( G ) 〉 ℤ = Λ ( G , ℤ ) is a free sub-module of Λ ( G , ℤ ) . Thus, it suffices to show (22) in order to prove the converse inclusion. We argue by induction on | V | − # G . If | V | − # G = 0 then there are only loops. Thus,

dim ℤ 〈 O el ( G ) 〉 ℤ = | E | = | E | − | V | + # G (23)

and the claim follows. Now assume that | V | − # G > 1 . Let f ∈ E be not a loop; then we consider the contracted graph G / f . Let c ∈ O el ( G ) if f ∉ E ( c ) and V ( c ) ∩ V ( f ) = ∅ or V ( f ) ∩ V ( c ) = { head ( f ) } , V ( f ) ∩ V ( c ) = { tail ( f ) } then certainly c / f ∈ O el ( G ) . If f ∈ E ( c ) then c / f remains elementary. If f ∈ E \ E ( c ) and V ( f ) ∩ V ( c ) = V ( f ) then c / f consists of two elementary cycles c 1 , c 2 ∈ O el ( G / f ) with one of them being a loop. Thus, G ( O el ( G / f ) ) = G ( O ¯ el ( G / f ) ) . By induction, we therefore compute

dim ℤ 〈 O el ( G / f ) 〉 ℤ = | E / f | − | V / f | + # G / f = | E | − | V | + # G ≥ dim ℤ 〈 O el ( G ) 〉 ℤ   . (24)

Step 2a: We show that dim ℤ 〈 O el ( G ) 〉 ℤ ≥ dim ℤ 〈 O el ( G / f ) 〉 ℤ . Let c ∈ O el ( G ) with f ∈ E ( c ) . By choosing an equivalent cycle c ¯ , c ∼ c ¯ if necessary w.l.o.g. we can assume that c is ordered such that c = ( e 1 , ⋯ , e | c | − 1 , f ) ∈ O el ( G ) , e i ∈ E , 1 ≤ i < | c | . Conversely, for every cycle d ∈ O el ( G / f ) we choose an ordering d = ( e 1 , ⋯ , e | c ′ | ) such that either d ˜ = ( e 1 , ⋯ , e | c ′ | , f ) or d ˜ = ( e 1 , ⋯ , e | c ′ | ) becomes a cycle in O ¯ el ( G ) . We consider the map

τ : O el ( G / f ) ↪ O ¯ el ( G ) ,     with     τ ( d ) : = d ˜   . (25)

Given h 1 , ⋯ , h l ∈ O el ( G / f ) , l ∈ ℕ such that [ h 1 ] , ⋯ , [ h l ] ∈ E ( G / f ) are a basis of 〈 O el ( G / f ) 〉 ℤ . Then τ induces a ℤ -linear injective lift

τ * : 〈 O el ( G / f ) 〉 ℤ → 〈 O ¯ el ( G ) 〉 ℤ ,     τ * ( [ h i ] ) = [ τ ( h i ) ] = [ h ˜ i ]   ,       i = 1, ⋯ , l   . (26)

Step 2b: We show that im ( τ * ) ⊆ 〈 O el ( G ) 〉 ℤ . Indeed, if d ∈ O el ( G / f ) is such that [ d ˜ ] has no negative entries then d ˜ = τ ( d ) ∈ O el ( G ) and [ d ˜ ] = τ * ( [ d ] ) ∈ 〈 O el ( G ) 〉 ℤ . If this is not the case then the entry [ d ˜ ] f = − 1 corresponding to f is the only negative entry. Since G ( O el ( G ) ) = G ( O ¯ el ( G ) ) there is b ∈ O el ( G ) with f ∈ E ( b ) .

For any x , y ∈ V we denote by d x , y the directed subpath of d ˜ connecting x and y. Further, let x 1 , ⋯ , x n ∈ V ( d ˜ ) be all crossing vertices such that for the edges e i , f i ∈ E ( b ) with head ( e i ) = x i = tail ( f i ) we have e i ∉ E ( d ˜ ) and f i ∈ E ( d ˜ ) , i = 1 , ⋯ , n . Analogously, we consider all crossing vertices y 1 , ⋯ , y n ∈ V ( d ˜ ) such that for the edges h i , g i ∈ E ( b ) with head ( h i ) = y i = tail ( g i ) we have h i ∈ E ( d ˜ ) and g i ∉ E ( d ˜ ) . Moreover, we assume that tail ( f ) = x 1 , head ( f ) = y 1 and that the x i , y i are ordered w.r.t the ordering of directed path d x 1 , y 1 . We consider two cases.

Case 1: If n = 1 then setting d 0 = b , d 1 = b y 1 , x 1 ∗ d x 1 , y 1 yields d x 1 , y 1 = f , see Figure 2 and thereby

[ d ˜ ] + [ d 0 ] − [ d 1 ] = 0 (27)

implying [ d ˜ ] = τ * ( [ d ] ) ∈ 〈 O el ( G ) 〉 ℤ .

Case 2: If n > 1 then we construct three cycles d 0 , d 1 , d 2 ∈ O el ( G ) by

d 0 = b   ,   d 1 = d x 1 , y n ∗ b y n , x 1   ,   d 2 = b y 1 , y n ∗ d y n , y 1   , (28)

see again Figure 3 for an illustration, where d 0 is indicated by the black lines. Since b , d ˜ ∈ O el ( G ) are elementary we have that the connected components of

b \ d ˜ = b y n , x 1 ∐ i = 1 , ⋯ , n − 1     b y i , x i + 1   ,   b ∩ d ˜ = ∐ i = 1 , ⋯ , n     d x i , y i (29)

are pairwisely disjoint paths. Moreover, d 0 , d 1 , d 2 ∈ O el ( G ) are elementary. Therefore, we verify again that

[ d ˜ ] + [ d 0 ] − [ d 1 ] − [ d 2 ] = 0 (30)

holds and thereby implies that [ d ˜ ] = τ * ( [ d ] ) ∈ 〈 O el ( G ) 〉 ℤ . Hence, im ( τ * ) ⊆ 〈 O el ( G ) 〉 ℤ and we have proven Step 2. Thus, due to Corollary 1 we have proven the theorem for R = ℤ .

Step 3: For R = ℤ 2 the claim follows by using that G ( O el ( G ) ) = G ( O ¯ el ( G ) ) is equivalent to 〈 O ¯ el ( G ) 〉 ℤ = 〈 O el ( G ) 〉 ℤ as proven above. Then the identity

Λ ( G , ℤ 2 ) ≅ 〈 O ¯ el ( G ) 〉 ℤ / 2 E ( G ) = 〈 O el ( G ) 〉 ℤ / 2 E ( G ) ≅ 〈 O el ( G ) 〉 ℤ 2 (31)

proves the theorem with respect to ℤ 2 -coefficients.

Remark 3. For simplification, the sketch of Case 2 in Figure 3 assumes that the paths b x i , y i are not nested, which might be not true in general. However, this circumstance does not affect (29) nor the fact that d 0 , d 1 , d 2 are elementary, which were all assumptions we used for the argumentation.

An immediate consequence of Theorem 5 is the following.

Corollary 2. Let G = ( V , E , head , tail ) be a multi-digraph. Then

dim R 〈 O el ( G ) 〉 R = dim R Λ ( G ( O el ( G ) ) , R ) ,   R = ℤ , ℤ 2 (32)

can be determined in O ( | E | ) .

Proof. By Theorem 5 we have 〈 O el ( G ) 〉 R = Λ ( G ( O el ( G ) ) , R ) and therefore dim R 〈 O el ( G ) 〉 R = dim R Λ ( G ( O el ( G ) ) , R ) . Now G ( O el ( G ) ) is given by the union of all strongly connected components of G which can be computed in linear time [16] yielding the claim due to (22).

The opportunity of computing the dimension of the directed cycles space 〈 O el ( G ) 〉 R efficiently, potentially addresses many challenging problems in graph theory. We use the result to study the structure of G in terms of cellular homology groups.

An excellent introduction to algebraic topology is given by Allen Hatcher [15].

Especially, the notion of CW complexes, geometric realizations and the classic theory of simplicial and cellular homology were presented explicitly therein. The following notions and results take these concepts for granted.

Definition 5 (Coherent Orientation). Let G = ( V , E , head , tail ) be a multi-digraph and O ⊆ O ¯ el ( G ) be a set of elementary connected cycles. A map o : E → { − 1,0,1 } is called a coherent orientation w.r.t. O iff

i) o ( e ) = 0 for all e ∈ E \ E ( O )

ii) [ c ] = ∑ i = 1 n     x i e i = ± ∑ i = 1 n     o ( e i ) e i for all characteristic vectors [ c ] with c ∈ O .

Lemma 6. Let G = ( V , E , head , tail ) be a multi-digraph and O ⊆ O ¯ el ( G ) be a set of elementary connected cycles. Then there exists a coherent orientation o : E → { − 1,0,1 } .

Proof. We argue by induction on | O | . If | O | = 1 then we choose an ordering c = ( e 1 , ⋯ , e n ) ∈ O and thereby a representative of all equivalent cycles in O. Thus, setting o ( e i ) = x i with [ c ] = ∑     x i e i defines a coherent orientation. Now assume that | O | > 1 . Let c = ( e 1 , ⋯ , e n ) ∈ O and O ′ ⊊ O be the set of all cycles being not equivalent to c. We consider G ′ = G ( O ′ ) and a splitting G ′ = G 1 ∪ ⋯ ∪ G K into elementary connected components. That is

∪ i = 1 K     O ∩ O ¯ el ( G i ) = O ′         and         O ′ ∩ O ¯ el ( G i ) ∩ O ¯ el ( G j ) = ∅   ,       ∀ 1 ≤ i < j ≤ K   . (33)

By induction there is a coherent orientation o ′ w.r.t. O ′ . Let [ c ] = ∑ i = 1 n     x i e i be the characteristic vector of c. Assume there are 1 ≤ k , l ≤ n with o ′ ( e k ) = x k and o ′ ( e l ) = − x l . Then e k , e l belong to different components, i.e., e k ∈ E ( G i ) , e l ∈ E ( G j ) , i ≠ j . Observe, that we can change o ′ to − o ′ on every component G i by keeping the requirements of a coherent orientation preserved. Thus, we can change o ′ to o ˜ such that o ˜ ( e i ) = x i for all e i ∈ E ( c ) ∩ E ( O ′ ) . Hence, setting o = o ˜ on E ( O ′ ) and o ( e i ) = x i for all e i ∉ E ( O ′ ) yields a coherent orientation.

Definition 6 (Elementary CW Complex). Let G = ( V , E , head , tail ) be a multi-digraph and X 1 ( G ) be its geometric realization. Further, let O ⊆ O ¯ el ( G ) be a set of elementary connected cycles. The geometric realization X ( G , O ) given by attaching 2-cells to X 1 ( G ) accordingly to O is called the elementary CW complex of G w.r.t. O. Further, X ¯ 0 , X ¯ 1 , X ¯ 2 shall denote the set of 0,1,2 -cells of X ( G , O ) , and

L 0 : X ¯ 0 → V   ,   L 1 : X ¯ 1 → E   ,   L 2 : X ¯ 2 → O (34)

the maps yielding the identification of 0,1,2 -cells with vertices, edges and cycles, respectively.

Indeed, for the complexes X ( G , O el ( G ) ) , X ( G , O ¯ el ( G ) ) the cellular homology can be derived by considering the following chain complexes.

Definition 7 (Elementary Chain Complex). Let G be a multi-digraph, O ⊆ O ¯ el ( G ) , be a set of elementary connected cycles, R = ℤ , ℤ 2 and o : E → { − 1,0,1 } be a coherent orientation w.r.t. O. Let X ( G , O ) be the associated CW complex. We consider the free generated R-modules

C 2 = ⊕ σ 2 ∈ X ¯ 2 ( G , O )     σ 2 R   ,   C 1 = ⊕ σ 1 ∈ X ¯ 1 ( G )     σ 1 R   ,   C 0 = ⊕ σ 0 ∈ X ¯ 0 ( G ) σ 0 R   . (35)

For R = ℤ , we define the boundary operators ∂ k : C k → C k − 1 , k = 1 , 2 on the generators by ∂ 2 ( σ 2 ) = ∑ σ 1 ∈ X ¯ 1 ( G )     m ( σ 2 , σ 1 ) σ 1 ,

m ( σ 2 , σ 1 ) = { o ( L 1 ( σ 1 ) )   , if       L 1 ( σ 1 ) ⊆ E ( L 2 ( σ 2 ) ) 0   , else (36)

and ∂ 1 ( σ 1 ) = ∑ σ 0 ∈ X 0     n ( σ 1 , σ 0 ) σ 0 ,

n ( σ 1 , σ 0 ) = { 1   , if       head ( L 1 ( σ 1 ) ) = L 0 ( σ 0 ) − 1   , if       tail ( L 1 ( σ 1 ) ) = L 0 ( σ 0 ) 0   , else (37)

For R = ℤ 2 we consider m ( σ 2 , σ 1 ) mod 2 ℤ and n ( σ 1 , σ 0 ) mod 2 ℤ , respectively. We enforce ∂ k to be R-linear by setting m ( λ σ 2 + μ σ ¯ 2 , σ 1 ) : = λ m ( σ 2 , σ 0 ) + μ m ( σ ¯ 2 , σ 0 ) , n ( λ σ 1 + μ σ ¯ 1 , σ 0 ) : = λ m ( σ 1 , σ 0 ) + μ n ( σ ¯ 1 , σ 0 ) for all σ 2 , σ ¯ 2 ∈ X ¯ 2 ( G , O ) , σ 1 , σ ¯ 1 ∈ X ¯ 1 ( G ) , σ 0 ∈ X ¯ 0 ( G ) , λ , μ ∈ R . Finally, we set C 3 , C − 1 = 0 and ∂ k = 0 for k ∈ { 0,3 } .

Remark 1 asserts why even in case of ℤ -coefficients the maps ∂ k yield well defined boundary operators. More precisely:

Theorem 7. Let G = ( V , E , head , tail ) be a multi-digraph, O ⊆ O ¯ el ( G ) be a set of connected elementary cycles and o : E → { − 1,0,1 } be a coherent orientation w.r.t. O. Let X ( G , O ) be the elementary CW complex, R = ℤ , ℤ 2 and ( C k , ∂ k ) be given due to Definition 7.

i) The pairs ( C k , ∂ k ) define a finite chain complex, i.e., ∂ k − 1 ∘ ∂ k = 0 for all k = 0 , 1 , 2 , 3 .

ii) The cellular homology groups H k ( X ( G , O ) , R ) are free R-modules H k ( X ( G , O ) , R ) : = ker ( ∂ k ) / im ( ∂ k + 1 ) , k = 0 , 1 , 2

H 2 ( X ( G , O ) , R ) ≅ R b 2 ( G , O , R ) , (38)

H 1 ( X ( G , O ) , R ) ≅ R b 1 ( G , O , R ) , (39)

H 0 ( X ( G , O ) , R ) ≅ R b 0 ( G , O , R ) . (40)

iii) The Betti numbers are given by

b 2 ( G , O , R ) = | O | − dim R 〈 O 〉 R (41)

b 1 ( G , O , R ) = | E | − | V | + # G − dim R 〈 O 〉 R (42)

b 0 ( G , O , R ) = # G   . (43)

Proof. By considering the sequence

0 → ∂ 3 C 2 → ∂ 2 C 1 → ∂ 1 C 0 → ∂ 0 0 (44)

the identities ∂ k − 1 ∘ ∂ k = 0 , follow for k ∈ { 1,3 } . For γ = ∑ k = 1 n     λ k σ k 1 ∈ C 1 and δ = ∑ k = 1 m     α k σ k 0 ∈ C 0 we consider the induced characteristic vectors

[ γ ] = ∑ k = 1 n     λ k [ L 1 ( σ k 1 ) ] ,   [ γ ] 2 = ∑ k = 1 n     μ k [ L 1 ( σ k 1 ) ] 2 ,       μ k = λ k mod 2   . (45)

[ δ ] = ∑ k = 1 m     α k [ L 0 ( σ k 0 ) ] ,   [ δ ] 2 = ∑ k = 1 m     β k [ L 0 ( σ k 0 ) ] 2   ,       β k = α k mod 2. (46)

Then we observe that [ ∂ 1 γ ] = I ( G , ℤ ) [ γ ] , [ ∂ 1 γ ] 2 = I ( G , ℤ 2 ) [ γ ] 2 , where I ( G , R ) denotes the incidence operator of G. Thus, ∂ 1 γ = 0 if and only if γ corresponds to a cycle, i.e., [ γ ] ∈ Λ ( G , ℤ ) , [ γ ] 2 ∈ Λ ( G , ℤ 2 ) . By definition of ∂ 2 and Lemma 3, this yields im ( ∂ 2 ) ⊆ ker ( ∂ 1 ) ≅ ker I ( G , R ) and consequently ∂ 1 ∘ ∂ 2 = 0 , showing (i). Furthermore, we realize that, im ( ∂ 3 ) = 0 and im ( ∂ 2 ) ≅ 〈 O 〉 R . Thus, im ( ∂ 2 ) is free generated by O, implying that im ( ∂ 2 ) ≅ R dim R 〈 O 〉 R . By using the dimension formula we realize that the quotient H 2 ( X ( G , O ) , R ) = C 2 / im ( ∂ 3 ) is a free R-module of dimension

b 2 ( G , O , R ) = | O | − dim R 〈 O 〉 R . (47)

Likewise, ker ( ∂ 1 ) ≅ ker I ( G , R ) implies that H 1 ( X ( G , O ) , R ) ≅ Λ ( G , R ) / 〈 O 〉 R . Since 〈 O 〉 R and Λ ( G , R ) are free R-modules, there are isomorphisms ρ : 〈 O 〉 R → R dim R 〈 O 〉 R , τ : Λ ( G , R ) → R | E | − | V | + # G . Let i : R dim R 〈 O 〉 R ↪ R | E | − | V | + # G be the natural embedding and B O be a basis of 〈 O 〉 R then B ′ O = i ∘ ρ ( B O ) can be extended to a basis B R of R | E | − | V | + # G and τ − 1 ( B R ) yields a basis of Λ ( G , R ) containing B O as a sub-basis of 〈 O 〉 R . Thus,

R | E | − | V | + # G / R dim R 〈 O 〉 R = τ ( Λ ( G , R ) ) / ( i ∘ ρ ( 〈 O 〉 R ) ) ≅ Λ ( G , R ) / 〈 O 〉 R ,

showing that the 1-st homology H 1 ( X ( G , O ) , R ) is a free R-module of dimension

b 1 ( G , O , R ) = dim R ( Λ ( G , R ) / 〈 O 〉 R ) = | E | − | V | + # G − dim R 〈 O 〉 R . (48)

Since C 1 and ker ( ∂ 1 ) are free R-modules we analogously observe that there is a sub-basis of ker ( ∂ 0 ) generating im ( ∂ 1 ) . Hence, H 0 ( X ( G , O ) , R ) is a free R-module of dimension

b 0 ( G , O , R ) = | V | − dim R im ( ∂ 1 ) = | V | − ( | E | − dim R ker ( ∂ 1 ) ) = | V | − ( | E | − dim R Λ ( G , R ) ) = # G   , (49)

which proves (iii).

Remark 4. Note that the Betti numbers do not depend on the choice of the coherent orientation o. Even though, Lemma 6 provides the existence of a coherent orientation o an explicit construction of o, which might be computational expensive, is not necessary to compute the b k ( G , O , ℤ ) , k = 0 , 1 , 2 .

Remark 5. The singular homology H n sing ( X ) of any CW complex X is isomorphic to its cellular homology [15] (Theorem 2.35). Thus, Theorem 7 provides a path for computing the singular homology of the elementary CW complexes X ( G , O ) . Further, we obtain the following consequence that yields main Theorem 2.

Corollary 3. Let the assumptions of Theorem 7 be fulfilled and denote with b k the R-rank of the cellular homology group H k ( X ( G , O ) ) , k = 0 , 1 , 2 .

i) b 0 ( G , O el ( G ) , R ) = b 0 ( G , O ¯ el ( G ) , R ) = # G can be computed in O ( | V | 2 + | V | | E | ) .

ii) b 1 ( G , O ¯ el ( G ) , R ) = 0 and b 1 ( G , O el ( G ) , R ) can be computed in O ( | V | 2 + | V | | E | ) .

iii) Computing b 2 ( G , O el ( G ) , R ) , b 2 ( G , O ¯ el ( G ) , R ) is #P-hard (sharp P-hard).

Proof. The number of connected components of any multi-graph can be determined by breadth-first search within O ( | V | 2 + | V | | E | ) [18], yielding i) due to Theorem 7. To show ii), recall that by Lemma 3, we have that 〈 O ¯ el ( G ) 〉 R = Λ ( G , R ) yielding b 1 ( G , O ¯ el ( G ) , R ) = 0 . Due to Corollary 2, computing dim R 〈 O el ( G ) 〉 R can be done in O ( | E | ) . Thus, by Theorem 7, the bottleneck for computing b 1 ( G , O , R ) is again given by determing #G yielding ii). Now iii) is a consequence of Theorem 7 and the fact that, by reduction to the Hamiltonian cycle problem, counting the number of all directed (connected) elementary cycles of a digraph is #P-hard [12].

We recall that two multi-digraphs G 1 = ( V 1 , E 1 ) and G 2 = ( V 2 , E 2 ) are called homeomorphic if there are sets of directed paths P 1 , P 2 such that deg ( u ) = 2 for any inner vertex u ∈ V ( p ) of p ∈ P 1 or p ∈ P 2 , respectively, and the graphs G 1 / P 1 ≅ G 2 / P 2 given by contracting P 1 , P 2 are isomorphic.

Corollary 4. Let G 1 and G 2 be two homeomorphic multi-digraphs. We denote with O i = O el ( G i ) , O ¯ i = O ¯ el ( G i ) , consider the elementary CW complexes X ( G i , O i ) , X ( G i , O ¯ i ) , i = 1 , 2 and R = ℤ , ℤ 2 . Then

H k ( X ( G 1 , O 1 ) , R ) ≅ H k ( X ( G 2 , O 2 ) , R ) , (50)

H k ( X ( G 1 , O ¯ 1 , R ) ) ≅ H k ( X ( G 2 , O ¯ 2 , R ) ) , (51)

where k = 0 , 1 , 2 .

Proof. As one can readily verify, neither the number of cycles | O el ( G i ) | , | O ¯ el ( G i ) | nor the dimension dim R 〈 O el ( G i ) 〉 R , 〈 O ¯ el ( G i ) 〉 R nor the number of connected components of G i change under the contraction of the paths P i , i = 1 , 2 yielding the claim due to Theorem 7.

Due to Corollary 3 (3) counting | O el ( G ) | , | O ¯ el ( G ) | is #P-hard. Though there are efficient algorithms [19] counting the number of cycles approximately, we, here, suggest to consider certain sub-fillings O ⊆ O el ( G ) , O ¯ el ( G ) to yield an suitable setup.

In this section we discuss several possibilities of filling the graph G by choosing O ⊊ O ¯ el ( G ) , such that the cellular homology of the resulting CW complexes X ( G , O ) can be computed efficiently and the filling is invariant under automorphisms, i.e., is independent of the chosen representation of G. The list below is by no means complete and just provides some suggestions.

We expect the concept of studying a graph G by computing its cellular homology groups with respect to specified fillings O ⊆ O ¯ el ( G ) to open up many possible applications. A short, non-exclusive list of ideas is presented below.

We extended the classic Theorem of Biggs to the case of directed cycles of multi-digraphs. These findigs were then incorporated to extend the classic interpretation of a graph G being a 1-dimensional CW complex to the 2-dimensional case by attaching 2-cells accordingly to an a priori specified set of elementary cycles O ⊆ O ¯ el ( G ) . We proved the associated cellular homology groups H k ( G , O , R ) , k = 0 , 1 , 2 to be free R-modules and derived formulas for computing their Betti numbers. Of course, one could ask for extending these considerations to even higher dimensions. However, in order to describe a 3-cell, we need to know all cycles forming its boundary. Thus, dealing with higher-dimensional cells makes computing the Betti numbers combinatorically harder, which made us restricting to the presented case.

The authors declare no conflicts of interest regarding the publication of this paper.

Hecht, M. and Sbalzarini, I.F. (2021) Biggs Theorem for Directed Cycles and Topological Invariants of Digraphs. Advances in Pure Mathematics, 11, 573-594. https://doi.org/10.4236/apm.2021.116037