Characteristic flows on signed graphs and short circuit covers

We generalise to signed graphs a classical result of Tutte [Canad. J. Math. 8 (1956), 13--28] stating that every integer flow can be expressed as a sum of characteristic flows of circuits. In our generalisation, the r\^ole of circuits is taken over by signed circuits of a signed graph which occur in two types -- either balanced circuits or pairs of disjoint unbalanced circuits connected with a path intersecting them only at its ends. As an application of this result we show that a signed graph $G$ admitting a nowhere-zero $k$-flow has a covering with signed circuits of total length at most $2(k-1)|E(G)|$.


Introduction
It is well known that every integer flow on a graph can be expressed as a sum of characteristic flows of circuits. By the characteristic flow χ C of a circuit C in a graph G we mean the flow that takes values +1 or −1 on C and value 0 anywhere else in G. One way of seeing this fact is to fix an orientation of G, take an arbitrary spanning tree T of G, and express the given flow φ as where x is a cotree edge, T (x) ⊆ T + x is the fundamental cycle corresponding to x, and the sum extends over all cotree edges of G. By choosing appropriate orientations for the cotree edges one can achieve that the coefficients φ(x) in this sum are all non-negative. Note that this choice induces an orientation of each fundamental cycle T (x) of G and may cause that an edge t of T belonging to two fundamental circuits T (y) and T (z) receives two opposite orientations from them; equivalently, for the same direction of t one would have χ T (y) (t) = +1 and χ T (z) (t) = −1. It turns out, however, that incompatibilities such as this one can always be avoided by choosing the set of circuits properly. Indeed, in 1956 Tutte [11, 6.2] proved that the decomposition of φ into characteristic flows can always be performed in such a way that all the circuits occurring in the expression are directed circuits with respect to a suitable fixed orientation of G. The aim of this paper is to establish a similar result for flows on signed graphs using the concept of a signed circuit and its characteristic flow, which we explain in Section 2 and Section 3, respectively.
Our main result reads as follows.
Theorem. For every integer flow φ on a signed graph G there exists a set C of signed circuits of G which are consistently directed with respect to a suitable orientation of G and positive integers n C such that φ = C∈C n C χ C .
Rather than following Tutte's approach based on the use of chain groups we prove our theorem directly by employing a purely graph theoretical approach that inspects circuits and their signed analogues. Of course, the special case of balanced signed graphs will yield a proof of the original Tutte's result.
Our paper is divided into four sections. Section 2 reviews the basic concepts of signed graph theory with the emphasis on the notions related to flows. In Section 3 we introduce the concept of a characteristic flow of a signed circuit and prove our main result. In the final section we apply the main result to the study of the signed analogue of the shortest circuit cover problem recently initiated in [7].

Signed graphs and flows
A signed graph is a graph in which each edge is labelled with a sign, + or −. An orientation, or a bidirection, of a signed graph is obtained by dividing each edge into two half-edges and by assigning individual orientations to them subject to the following compatibility rule: a positive edge has one half-edge directed from and the other halfedge directed to its end-vertex, while a negative edge has both half-edges directed either towards or from their respective end-vertices. Thus each edge, irrespectively of its sign, has two possible orientations which are opposite to each other.
Given an abelian group A, an A-flow on a signed graph G is an assignment of an orientation and a value from A to each edge in such a way that for each vertex of G the sum of incoming values equals the sum of outgoing values (Kirchhoff 's law ). If 0 ∈ A is not used as a flow value, the flow is said to be nowhere-zero. The concept of a nowherezero A-flow is particularly interesting when A is the group of integers. A major problem is to determine, for a given signed graph G, the smallest integer k ≥ 2 such that G has an integer flow with values in the set {±1, ±2, . . . , ±(k − 1)}; such a flow is called a nowhere-zero k-flow. In 1983, Bouchet [3] conjectured that every signed graph that admits a nowhere-zero integer flow has a nowhere-zero 6-flow. Although various approximations of this conjecture have been proved [4,10,12,13,15], this conjecture remains open.
Signed graphs that admit a nowhere-zero integer flow are called flow-admissible. In contrast to unsigned graphs, describing flow-admissible is not immediate. For this purpose we need the notions of a balance of a signed graph and that of a signed circuit.
A circuit of a signed graph G is called balanced if it contains even number of negative edges, otherwise it is called unbalanced. A signed graph itself is called balanced if it does not contain any unbalanced circuit, and is called unbalanced if it does. The collection of all balanced circuits is the most fundamental characteristic of a signed graph: signed graphs having the same underlying graphs and the same sets of balanced circuits are considered to be identical, irrespectively of their actual signatures.
A signed circuit of a signed graph is a subgraph of any of the following three types: (1) a balanced circuit, (2) the union of two disjoint unbalanced circuits with a path that meets the circuits only at its ends, or (3) the union of two unbalanced circuits that meet at a single vertex.
A signed circuit falling under item (2) or (3) is called an unbalanced bicircuit. Note that a bicircuit from item (3) can be regarded as as special case of the one from (2), with the connecting path being trivial. Observe, however, that signed circuits from items (1) or (3) admit a nowhere-zero 2-flow while those under item (2) admit a nowhere-zero 3-flow, but not a 2-flow.
The following result is due to Bouchet [3] and reflects the fact that signed circuits are inclusion minimal signed graphs that admit a nowhere-zero integer flow.

Decomposition into characteristic flows
Let φ be a flow on a signed graph G. If we reverse the orientation of any edge e and replace the value φ(e) with −φ(e), the resulting valuation will again be a flow. We regard this operation as a way of expressing the same flow φ in terms of a different orientation. Thus, within a given signature, we may choose a compatible orientation arbitrarily. If φ is an integer flow, we can always find an orientation for G such that φ(e) ≥ 0 for each edge e. We call this orientation a positive orientation of G with respect to φ. If φ is nowhere-zero, this orientation is unique.
Another useful operation that preserves flows on a signed graph is known as switching. It consist in choosing a vertex v of G, reversing the orientation of each half-edge incident with v, and changing the signature of G accordingly. The result is an identical signed graph, because the total sign of every circuit has not been changed, equipped with a new compatible orientation. If G carries a flow, then the same function works as a flow for the new signature and orientation. The same flow is thus again expressed in terms of a different orientation and signature. By a repeated use of switching we may turn a given signature into any other equivalent signature [14], keeping the flow invariant.
For a fixed orientation of G, the sum φ + ψ of two flows φ and ψ is defined by setting (φ + ψ)(e) = φ(e) + φ(e); clearly, φ + ψ is again a flow. We are now interested in the reverse process of expressing an arbitrary integer flow as a sum of suitably chosen elementary flows. The question whether this is possible for every flow on an arbitrary flowadmissible signed graph was posed by André Raspaud (personal communication) referring to a result of Tutte [11, 6.2] for unsigned graphs. In Tutte's theorem, elementary flows are represented by characteristic flows of circuits. Theorem 2.1 suggests that in the case of flows on signed graph circuits should be replaced with signed circuits.
Consider a pair of adjacent edges e and f sharing a vertex v in a bidirected signed graph. We say that the walk ef is consistently directed at v if exactly one of the two halfedges incident with v is directed to v. A path or a balanced circuit is said to be consistently directed if all pairs of consecutive edges are consistently directed. An unbalanced circuit is consistently directed if it has a single vertex, called the faulty vertex, such that all pairs of consecutive edges are consistently directed except for the pair sharing the faulty vertex. Finally, an unbalanced bicircuit is said to be consistently directed if every pair of adjacent edges in the bicircuit is consistently directed except for the two edges of either circuit that share an end-vertex of the connecting path. These vertices are the faulty vertices of the bicircuit. Examples of consistently directed signed circuits are displayed in Fig. 1. It is easy to see that any signed circuit in a bidirected signed graph can be turned into a consistently directed signed circuit by only reversing the orientation edges. Furthermore, if the signature is switched at some vertex, consistency of orientation is not affected. Hence, any signed circuit may have several different consistent orientations.
Definition. Let G be a signed graph and let C be a signed circuit of G endowed with a consistent orientation. The characteristic flow of C in G is a function χ C : E(G) → {0, 1/2, 1} defined as follows. If C is a balanced circuit, we set χ C (e) = 1 for each edge of S and χ S (e) = 0 otherwise. If C is an unbalanced bicircuit, we set χ C (e) = 1 whenever e belongs to the connecting path of C, χ C (e) = 1/2 whenever e belongs to a circuit of C, and χ C (e) = 0 otherwise.
Note that the characteristic flow χ C of a signed circuit C is a flow although not necessarily an integer flow. The values of χ C and, in fact, the values of an arbitrary linear combination of characteristic flows of signed circuits of a graph over integers will be contained in the cyclic subgroup H ≤ Q generated by the element 1/2. The group H includes the group of integers as a subgroup of index 2; the elements of H − Z will be called fractional.
Since a flow on a signed graph is invariant under the orientation reversal and vertexswitching, the concept of a characteristic flow applies to any signed circuit irrespectively of its particular orientation and signature. Furthermore, it is easy to see that, up to equivalence, the characteristic flow of a signed circuit is uniquely determined by the value on a single bidirected edge, which may be either +1 or −1. However, switching at all vertices reverses the orientation of each edge without changing the flow values. This implies that, up to equivalence, every flow φ on a signed graph coincides with its opposite −φ. In particular, each signed circuit has exactly one characteristic flow, up to equivalence.
We proceed to the main result, the decomposition theorem. For the proof recall that the support of a flow φ, denoted by supp(φ), is the set of all edges e for which φ(e) = 0. Remark. A natural question arises whether for this theorem to be true the fractional values in the definition of a characteristic flow are really necessary. The answer is, unfortunately, 'yes'. To see this, let us consider the signed graph G consisting of two vertices joined by a pair of positive parallel edges with a negative loop attached at either vertex. It is easy to see that G admits a nowhere-zero 2-flow, but any decomposition of this flow into the sum of characteristic flows will contain characteristic flows of two distinct unbalanced bicircuits, each with coefficient 1. The reader can easily extend this example into an infinite series of similar examples where a nowhere-zero 2-flow only decomposes into the sum of characteristic flows of two distinct unbalanced bicircuits, each with coefficient 1.
Proof of Theorem 3.1. Throughout the proof we keep fixed a positive orientation of G with respect to φ. The proof is trivial if φ = 0, so we may assume that supp(φ) = ∅. If supp(φ) contains a consistently directed balanced circuit B, we form the flow φ − χ B which is again an integer flow on G. We repeat the procedure with the flow φ − χ B and continue as long as the support of the current flow contains a consistently directed balanced circuit. Eventually we obtain a set B of consistently directed balanced circuits and the flow φ 1 = φ − B∈B χ B . If φ 1 = 0, then φ = B∈B χ B , and the required expression for φ follows immediately. Otherwise supp(φ 1 ) is nonempty and induces a subgraph G 1 which carries a non-null integer flow φ 1 . Clearly, G 1 contains no consistently directed balanced circuit.
Our next aim is to show that G 1 contains a consistently directed unbalanced bicircuit. To this end, we first identify a consistently directed unbalanced circuit in G 1 . We pick an arbitrary vertex u 1 of G 1 and choose an edge e 1 that leaves u 1 ; since G 1 has a positive orientation such an edge always exists. Let u 2 be the other end of e 1 . At u 2 , there must be an edge e 2 such that the walk e 1 e 2 is consistent at u 2 . We continue in the same manner until we reach a vertex previously visited, say u i . The segment between the two occurrences of u i is clearly a circuit D of G 1 . By the construction, D is consistent everywhere except possibly u i . Since G 1 contains no consistently directed balanced circuit, D is a consistently directed unbalanced circuit and u i is its faulty vertex.
Next we show that D is contained in a directed unbalanced bicircuit. The edges of D incident with u i are either both directed to u i or both from u i . Since G 1 has a positive orientation, there is an edge f 1 incident with u i which is consistent at u i with any of the two edges of D incident with u i . Set v 1 = u i and let v 2 denote the other end of f 1 . At v 2 , there must be an edge f 2 such that the walk f 1 f 2 is consistent at v 2 . Again, we continue similarly until we reach a vertex v j that either belongs to D or coincides with a vertex v m with m < j. Observe that v j does not lie on D − v 1 , for if it does, we can divide D into two v j -v 1 -segments D 1 and D 2 exactly one of which is consistently directed with f j−1 at v j . But then one of f 1 f 2 . . . f j−1 D 1 or f 1 f 2 . . . f j−1 D 2 is a consistently directed balanced circuit, a contradiction (see Fig. 2 for illustration). It follows that v j = v m for some m < j. In this case, however, D = f m f m+1 . . . f j−1 is a consistently directed unbalanced circuit which together with D and the path f 1 f 2 . . . f m−1 forms a consistently directed unbalanced bicircuit U . It may happen that u m = v i in which case the connecting path of the bicircuit is trivial.
We now take the characteristic flow χ U , construct the flow φ 2 = φ 1 − χ U , and set G 2 = supp(φ 2 ). Note that φ 2 is not an integer flow anymore, but its fractional values are confined to two edge-disjoint consistently directed unbalanced circuits of G 2 . Furthermore, G 2 contains no consistently directed balanced circuit because such a circuit would also be contained in G 1 , which is impossible. The next step is to show that G 2 contains a consistently directed unbalanced bicircuit U such that the support of the flow φ 3 = φ 2 − χ U has either none or exactly two edge-disjoint consistently directed unbalanced circuits carrying fractional values. Since neither supp(φ 2 ) nor supp(φ 3 ) contain consistently directed balanced circuits, repeating this procedure will necessarily terminate with a set U of consistently directed unbalanced bicircuits such that φ 1 − U ∈U χ U = 0, Figure 2: Contradiction in constructing an unbalanced bicircuit Since the latter expression immediately yields the statement of the theorem, all that remains is to describe a procedure that starts with a flow ψ on G such that (1) supp(ψ) contains no consistently directed balanced circuit, and (2) fractional values of ψ occur in exactly two edge-disjoint unbalanced circuits of supp(ψ).
and constructs an unbalanced bicircuit U ⊆ supp(ψ) such that in supp(ψ − χ U ) there are either none or exactly two edge-disjoint consistently directed unbalanced circuits carrying fractional values. Let ψ be a flow on G such that supp(ψ) satisfies (1) and (2) stated above. Let d and d be the faulty vertices of D and D , respectively. To construct an unbalanced bicircuit U let us take the circuit D, choose an edge e incident with d which is consistently directed with either of the two adjacent edges of D and proceed by successively constructing a consistently directed trail T until we either reach a previously encountered vertex of T or a vertex of D ∪ D . Let t be the terminal vertex of T .
First observe that t cannot belong to D − d. Indeed, otherwise we could split D into two t-d-segments D 1 and D 2 producing circuits T D 1 and T D 2 one of which would be a consistently directed balanced circuit in supp(ψ), contradicting (2).
There remain four possibilities for the position of t to consider.  Starting from d construct a consistently directed trail T whose first edge is consistently directed with either of the two adjacent edges of D and continue until we either reach a previously visited vertex of T or a vertex of D ∪ D ∪ T . Let t be the first such vertex.
Observe that t belongs neither to D − d not to D − d . Otherwise, in the former case D would contain a t -d-segment S such that ST D 1 T is a consistently directed balanced circuit in supp(ψ), and similarly in the latter case D would contain a t -d -segment S such that S T is a a consistently directed balanced circuit in supp(ψ). In both cases we would get a contradiction.
There remain two possibilities for the position of t .
Subcase 2.1. The vertex t coincides with a previously encountered vertex of T , possibly t = d . The portion of T between the two occurrences of t forms a consistently directed unbalanced circuit, say D , and hence D ∪ T is a consistently directed unbalanced bicircuit. If we set U = D ∪ T , then the support of the flow ψ − χ U again contains precisely two directed unbalanced circuits carrying fractional values, namely D and D .
Subcase 2.2. The vertex t belongs to T , possibly t = d. It is obvious that t = t because otherwise t would lie in D −d , which we have shown to be impossible. On the other hand, t may coincide with d. The vertex t splits the path T into two segments, a d-t -segment W 1 , which may be trivial, and a t -t-segment W 2 , which is nontrivial. Consider the last edge g of T and the first edge h of the segment W 2 . If g was consistently directed with h at t , then W 2 D 1 T would be a consistently directed balanced circuit within supp(ψ), which is impossible. Thus g is not consistently directed with h at t . Since h is consistently directed at t with the last edge of W 1 , provided that t = d, or with both edges of D incident with d, provided that t = d, it follows that W 1 ∪T is a consistently directed path whose ends d and d are the only faulty vertices of D ∪ W 1 ∪ T ∪ D . Thus D ∪ W 1 ∪ T ∪ D is a consistently directed unbalanced bicircuit, the sought U . It is now easy to see that ψ−χ U is an integer flow whose support does not contain any consistently balanced circuit.
Case 3. The vertex t coincides with d and the terminal edge of T is consistently directed at d with either of the two adjacent edges of D . In this case D ∪ T ∪ D is a consistently directed unbalanced bicircuit. We can set U = D ∪ T ∪ D and observe that ψ − χ U is an integer flow whose support contains no consistently directed balanced circuit. Again, the required conclusion holds. As we have seen, in each case the procedure can either be continued or will terminate with the zero flow. The proof is complete.
Observe that the proof of Theorem 3.1 makes no use of the characterisation of flowadmissible graphs given in Theorem 2.1. Just on the contrary, Theorem 2.1 easily follows from Theorem 3.1. Proof. The forward implication is an immediate consequence of Theorem 3.1. For the converse, let C = {C 1 , C 2 , . . . , C r } be a set of signed circuits such that each edge of G belongs to a member of C. We first fix an arbitrary orientation of G; note that the elements of C need not be consistently directed with respect to this orientation. Now we can define the function ψ : E(G) → Z by setting It is easy to see that ψ is indeed a nowhere-zero integer flow on G.

Application to signed circuit covers
A circuit cover of an unsigned graph is a collection of circuits such that each edge of the graph belongs to at least one of the circuits. It is a standard problem to find, for a given graph, a circuit cover of minimum total length. It has been conjectured that every bridgeless graph G has a circuit cover of length at most 7|E(G)|/5 (Jaeger, private communication; independently [1]), but the best current general bound is 5|E(G)|/3 (see [1,2]). The 7/5-conjecture is particularly interesting for its relationship to other prominent conjectures in graph theory. For instance, its validity is implied by the Petersen flow conjecture (alternatively known as the Petersen colouring conjecture) of Jaeger [5,Section 7], while the conjecture itself implies the celebrated cycle double cover conjecture (see Raspaud [9] and Jamshy and Tarsi [6]).
A natural analogue of a circuit cover for signed graphs is the concept of a signed circuit cover introduced in [7]. It is a collection C of signed circuits of a signed graph G such that each edge of G is contained in at least one member of C. In [7] it was shown that every flow-admissible signed graph G has a signed circuit cover of total length at most 11|E(G)|.
We now apply our Theorem 3.1 to the shortest signed circuit problem. If a signed graph G admits a nowhere-zero integer k-flow φ, we can decompose it into the sum C∈C n C χ C of characteristic flows guaranteed by Theorem 3.1. Obviously, the set C provides a signed circuit cover of G. This cover yields the following bounds.
Corollary 4.1. If a signed graph G has a nowhere-zero flow φ, then G has a signed circuit cover such that each edge e belongs to at most 2|φ(e)| signed circuits.
Corollary 4.2. If a signed graph G admits a nowhere-zero k-flow, then it has a signed circuit cover of total length at most 2(k − 1)|E(G)|.
Observe that if Bouchet's 6-flow conjecture [3] is true, then the previous corollary implies that every flow-admissible signed graph G has a signed circuit cover of total length at most 10|E(G)|.