Biggs Theorem for Directed Cycles and Topological Invariants of Digraphs

We generalize Biggs Theorem to the case of directed cycles of multi-digraphs allowing to compute the dimension of the directed cycle space independently of the graph representation with linear runtime complexity. By considering two-dimensional CW complex of elementary cycles and deriving formulas for the Betti numbers of the associated cellular homology groups, we extend the list of representation independent topological inavariants measuring the graph structure. We prove the computation of the 2nd Betti number to be sharp #P hard in general and present specific representation invariant sub-fillings yielding efficiently computable homology groups. Finally, we suggest how to use the provided structural measures to shed new light on graph theoretical problems as graph embeddings, discrete Morse theory and graph clustering.


Introduction
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 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 ( ) where #G denotes the connected components of G.
However, Biggs Theorem is only known to hold for the space generated by

Statement of Contribution
We denote by Consequently, in case of equivalence, there holds Indeed the construction of the induced subgraph i) and can be computed in Using that cellular and singular homology are isomorphic [15] Consequently, the graph structure of G can be understood in terms of singular homology. In light of this fact, we revisit classic problems in graph theory as graph embeddings, discrete Morse theory and graph clustering from this new perspective.

Preleminaries
For rendering the introduced concepts of this article and the proof Theorem 1 consistent we introduce a definition of graphs that yields a notion of multi-digraphs without requiring multi-sets.
c c′ can coincide in their edge sets but differ in their orderings an elementary cycle is uniquely determined up to cyclic reordering of its edges. This makes elementary cycles the pivotal choice of our considerations. We

Outline
In Section 2, we provide all ingredients to generalize Bigss Theorem and give the proof in Section 3. In Section 4, the 2-dimensional elementary CW complex of G and its homology is studied. Section 5 considers representation invariant sub-fillings while Sections 6 and 7 suggest applications and yield a conclusion of our results.

The Module of Cycles
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.

( )
, , head, tail G V E = be a multi-digraph we introduce the free R-modules, are called the characteristic vectors of ε with respect to  -and 2  -coefficients.
are well defined.

 
-linear incidence operator operators, defined on the generators by By choosing an enumeration The notion of ( is given by setting the set of all characteristic vectors induced by cycles in O. The free generated cycle modules are given by Hence, proving the claim.
To characterize the cycle spaces we rephrase Biggs Theorem [14] for connected cycles of simple digraphs matching our setup and notation.
then the dimension of the connected cycle spaces are given by 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.

( )
, , head, tail G V E = be a multi-digraph. Then the dimension of the connected cycle spaces are given by where #G denotes the number of connected components of G. Proof. Indeed, each loop or multiple edge increases the dimension by 1. More , , Recursively applying this procedure ends up with a simple digraph Thus, due to Theorem 4 we compute  How to generalize the characterisation of cycle spaces in the directed case is provided in the next section.

Biggs Theorem for Directed Cycles
We prove a generalization, we introduce the concept of contracting edges in a multi-digraph as follows. u v u v x  and , / : gives the quotient of V with respect to The multi-digraph  , 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., Now we have all ingredients to state the first main theorem of this article.
Theorem 5 (Biggs Theorem for directed cycles). Let Consequently, in case of equivalence, there holds Proof. The proof splits into several steps: Step 1: We show Thus, every i e hast to be contained in at least one of them proving Step 2: We show Thus, it suffices to show (22) in order to prove the converse inclusion. We argue by induction on then there are only loops. Thus, and the claim follows. Now assume that then we consider the contracted graph Step 2a: We show that we can assume that c is ordered such that Step 2b: We show that For any , x y V ∈ we denote by , are pairwisely disjoint paths. Moreover, Therefore, we verify again that holds and thereby implies that τ ⊆  and we have proven Step 2. Thus, due to Corollary 1 we have proven the theorem for R =  .
Step 3: For 2 R =  the claim follows by using that as proven above. Then the identity proves the theorem with respect to 2  -coefficients. An immediate consequence of Theorem 5 is the following.
can be determined in Proof. By Theorem 5 we have 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 ( ) el R O G 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.

The CW Complex of Elementary Cycles
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.    , .
For R =  , we define the boundary operators , , R λ µ ∈ . Finally, we set 3 1 , Remark 1 asserts why even in case of  -coefficients the maps k ∂ yield well defined boundary operators. More precisely: i) The pairs ( ) define a finite chain complex, i.e., ii) The cellular homology groups Proof. By considering the sequence . Thus, by Theorem 7, the bottleneck for computing ( ) 1 , , b 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].
Proof. As one can readily verify, neither the number of cycles

Representation Invariant Sub-Fillings
In this section we discuss several possibilities of filling the graph G by choosing , 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.

Fillings Induced by Cycle Bases
. Thereby, fillings by R-linear independent cycles or basis yield only trivial topological information.

Fillings Induced by Shortest Cycles
Let ( ) , , head, tail G V E = be a multi-digraph. We consider the set of all shortest

( )
, G V E = be a multi-digraph and ∆ its maximum degree. i) Proof. Note that the all-shortest path problem, which is to find a shortest connected path between all pairs of vertices , u v V ∈ is very well studied.

Fillings Induced by Cliques
Then we consider a set of cycles have been determined.
Note that the general clique problem, i.e., finding a clique of given size n ∈  in a graph G, belongs to the list of classic NP-complete problems [25]. However, if the size n is fixed, finding cliques can be done efficiently in . Even better performs the Bron-Kerbosch algorithm, which lists all maximal cliques [26]. Many other contributions in regard of clique detecting algorithms were made, allowing us to find a non-trivial filling as described above efficiently.
Consequently, the homology groups

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

Graph Embeddings
We strengthen the classic notion of cellular graph embeddings as follows. Hence, an elementary embedding of G into S is a special case of a cellular embedding and therefore a stronger requirement on the embedding ρ . The following example makes this circumstance visible.
Example 2. Figure 4 illustrates embeddings of 3 Since S is a closed surface, we have ( ) 2 2 , , if S is non-orientable [15].  [28]. In general, the problem of finding a cellular embedding of G into a orientable surface S of minimal genus is NP-complete [29]. However, finding a cellular embedding into a surface S of maximal genus is polynomial time solvable [30]. Our strengthened notion of embeddings and the relaxation of S being allowed to be non-orientable might provides new insights into embedding

Graph Clustering
Many applications require comparing graphs or clustering a set of graphs  into subsets of "similar" graphs. The distance functions : D + × →    used to measure similarity between graphs often lead to NP-complete optimization problems and can only be approximated. For instance, the graph-edit distance edit : D + × →    [31] is a (pseudo) metric measuring the cost of deleting and inserting the minimum number of edges and vertices required to modify a graph 1 G into a graph 2 G . Determining this distance in general is an NP-hard problem, which is why additional assumptions and restrictions to special graph classes are usually made. We posit that any feasible clustering 1 , , n ⊆     of a given set of graphs  , i.e., i j = ∅    , i j ≠ , and 1 n i i= =    , must fulfill the following properties: a) Let 1 2 , G G ∈  be isomorphic or more generally homeomorphic, then 1 2 , G G belong to the same cluster, i.e., 1 2 , i G G ∈  for some 1 i n ≤ ≤ . b) Given 1 2 , G G ∈  , the problem of deciding whether 1 G and 2 G will belong to the same cluster is solvable in polynomial time.

Conclusion
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 interpre-

Classic Graph-Theoretical Problems
It is natural to ask the question how classic graph-theoretical problems behave for certain classes of graphs. Here we provided some methods to classify graphs with respect to certain aspects of their topology. The fillings presented in Section 5 might be a good starting point of investigating classic graph theoretical problems on graphs with bounded topological complexity in terms of the derived Betti numbers. For instance the Feedback Arc Set Problem, which is to delete as less edges as possible to make G acyclic, could profit from certain obstructions on the cycle topology, [33] [34], as derived in Section 5.2. Further, the understanding of cliques of a graph is crucial if one wants to determine its chromatic numbers [35]. Despite for coloring problems [8], the minimal genus of a graph plays an important role within the unresolved Graph Isomorphism Problem [36] or extensions of Whitney's Theorem [11].

Related Complexes and Homology Theories
In addition to the concepts presented here, there are multiple other possibilities of assigning complexes and (co)-homologies to a given graph G [37] [38] [39].
Translations of these theories in terms of our contribution are certainly of interest.