RESOLUTION OF INDECOMPOSABLE INTEGRAL FLOWS ON A SIGNED GRAPH

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1 RESOLUTION OF INDECOMPOSABLE INTEGRAL FLOWS ON A SIGNED GRAPH BEIFANG CHEN, JUE WANG, AND THOMAS ZASLAVSKY Abstract. It is well-known that each nonnegative integral flow of a directed graph can be decomposed into a sm of nonnegative graph circit flows, which cannot be frther decomposed into nonnegative integral sb-flows. This is eqivalent to saying that indecomposable flows of graphs are those graph circit flows. Trning from graphs to signed graphs, the indecomposable flows are mch richer than that of ordinary nsigned graphs. The present paper is to give a complete description of indecomposable flows of signed graphs from the viewpoint of resoltion of singlarities by introdcing covering graphs. A real flow (also known as circlation) on a graph or a signed graph (a graph with signed edges) is a real-valed fnction on oriented edges sch that the net inflow to each verte is zero. An integral flow is a flow whose vales are integers. There are many reasons to be interested in flows on graphs; an important one is their relationship to graph strctre throgh the analysis of (conformally) indecomposable flows, that is, integral flows that cannot be decomposed as the sm of two integral flows having the same sign on each edge (both 0 or both 0). It is well known, and an important observation in the theory of integral network flows, that the indecomposable flows are identical to the circit flows, which have vale on the edges of a graph circit (=cycle) and 0 on all other edges. Etending the theory of indecomposable integral flows to signed graphs, which is stdied in [7] by algorithmic method, led to a remarkable discovery that there are, besides the anticipated circit flows (which are already more complicated in signed graphs than in ordinary graphs), many strange indecomposable flows with elaborate strctre not describable by signed-graph circits. In this article we characterize this strctre again by the method of sign-labeled covering graphs, lifting each verte and each edge of a signed graph to two vertices and two edges respectively of a sign-labeled covering graph, and lifting each indecomposable flow to a simple cycle flow in the sign-labeled covering graph. We think of the lifting as a combinatorial analog of resoltion of singlarities in algebraic geometry. The strange indecomposable flows are singlar phenomena, which we resolve by lifting them (blowing p overlapped edges) to ordinary cycle flows in a covering graph. Comparing to the algorithmic approach in [7], the present paper hints a connection (at least conceptally) between graph theory and covering spaces of algebraic topology and resoltion of algebraic geometry. We believe that this connection is sefl to stdy gain graphs [4] that are more complicated than signed graphs. Date: Janary 9, Mathematics Sbject Classification. Primary 05C; Secondary 05C0, 05C, 05C38. Key words and phrases. Signed graph, sign-labeled covering graph, prime Elerian signed graph, Elerian cycle-tree, indecomposable integral flow. Research of the first athor was spported by RGC Competitive Earmarked Research Grants , , and 6008.

2 It is characterized that indecomposable integral flows of signed graphs are characteristic vectors of certain so-called Elerian cycle-trees (see Definition 9, Theorem 3 and Theorem 5 below). The properties of Elerian cycle-trees lead to the following half-integer scale conformal decomposition: Every nontrivial integral flow of an oriented signed graph can be conformally decomposed into a positive half-integer (perhaps integer bt not always integer) linear combination of signed-graph circit flows.. Graphs and signed graphs Graphs. A graph is a system G = (V, E), with verte set V and edge set E, sch that each edge E is associated with a mltiset End() of two vertices, called the end-vertices of ; the edge is called a link if the two vertices of End() are distinct, and is called a loop if the two vertices are identical. Let be an edge and End() = {, v}; we say that is incident with and v, or and v are adjacent by ; we write (, ) and (v, ). A tricky technical point is that this notation does not distingish the two end-vertices of a loop; we take an easy way ot by treating {(, ), (v, )} as a mltiset when = v. A walk of length n in a graph G = (V, E) is a pair W = (, ) of fnctions : {0,,..., n} V, : {,,..., n} E sch that the end-vertices of the edge (i) are the vertices (i ) and (i). We write i = (i), i = (i), and W = 0... n n n. A walk is said to be closed if n and v 0 = v n and open otherwise. A sbseqence of the form i i i i... j j j is called a sb-walk of W. A walk is called a trail if there are no repeating edges, ecept the initial and terminal vertices for closed walk (sch a trail is called a closed trail). A walk is called a path (or simple walk) if there are no repeating vertices (and sbseqently there are no repeating edges), ecept the initial and terminal vertices for closed walk (sch paths are called closed paths). As sal, we call a closed path a cycle. A graph circit is a cycle; the name comes from the fact that the cycles of a graph form the circits of the graphic matroid whose elements are the edges of the graph. An isthms (or ct-edge) is an edge whose deletion increases the nmber of connected components. A ct-verte is a verte whose deletion, with all incident edges, increases the nmber of components, or that spports a loop and is incident with at least one other edge. A block is a maimal sbgraph withot ct-vertices. Ths, a loop or isthms or isolated verte is a (trivial) block. We call blocks adjacent if they have a common verte (which is necessarily a ct-verte). An end-block is a block adjacent to eactly one other block. For the notions of graph theory that we didn t define, we refer to the books [, 3, 0] Signed graphs. A signed graph Σ = (V, E, σ) consist of an ordinary graph G = (V, E) and a fnction σ : E {, } (signs are mltiplied rather than added), called the sign fnction of Σ. The sign of a walk W = 0... n n in Σ is the prodct n σ(w ) = σ( i ). i= In particlar, a cycle has a sign, which is either positive or negative. A sbgraph or its edge set is said to be balanced if every cycle in the sbgraph (indced by the edge set) is positive.

3 A signed-graph circit in a signed graph is a sbgraph (or its edge set) of one of the following three types: (i) A positive cycle, said to be of Type I. (ii) A pair of negative cycles whose intersection is a single verte, said to be of Type II (also known as a contrabalanced tight handcff ). (iii) A pair of verte-disjoint negative cycles together with a simple path of positive length (called the circit path) that connects the two cycles and is internally disjoint from the cycles, said to be of Type III (also known as a contrabalanced loose handcff ). The signed-graph circits form the circits of a matroid on the edge set of the signed graph []. An ordinary nsigned graph can be considered as a signed graph whose all edges are positive. When all edges are positive, every cycle is positive and the only signed-graph circits are those of Type I, that is, the graph circits. Orientation. A bi-direction of a graph (a concept introdced by Edmonds [8]) is a fnction from the verte-edge pairs to the sign grop. One thinks of an edge at its one end-verte with either a positive sign as having an arrow directed away from the end-verte, or a negative sign as having an arrow directed towards the end-verte. Ths each edge is assigned two arrows, one at each of its end-vertices. Technically, a bi-direction may be described by a mlti-valed fnction ε : V E {, 0, } sch that (i) ε(, ) = 0 if End(), (ii) ε(, ) is a nonzero vale if is a link, (iii) if is a loop then ε(, ) is a mltiset of two (possibly identical) nonzero vales. An orientation of a signed graph Σ is a bi-direction ε on its ndergrond graph sch that for each edge with end-vertices, v (possibly is a loop so that = v), σ() = ε(, )ε(v, ), = v. Then a positive edge has two arrows in the same direction along ths indicating a direction along as in an ordinary directed graph. A negative edge has two opposite arrows that both point away from or both point towards the end-vertices. A signed graph together with an orientation is known as an oriented signed graph; see [5, 6, 3]. Let (Σ, ε) be an oriented signed graph throghot. A sorce in (Σ, ε) is a verte at which all edges are directed otwards, that is, ε(, ) = for all edges at. Conversely, if all edges at point towards, the verte is called a sink. A walk W = 0... n n n of length n on an oriented signed graph is said to be coherent at i if ε( i, i ) ε( i, i ) = 0, that is, the order of W follows the orientations of edges at the verte i. When W is a closed walk, we apply this definition to 0 by taking sbscripts modlo n, that is, ε( n, n ) ε( 0, ) = 0. An open walk is said to be directed in (Σ, ε) if it is coherent at its every internal verte. A closed walk is said to be directed in (Σ, ε) if it is coherent at every verte, inclding at the initial and terminal verte. A direction of W is a fnction ε W with vales either or, defined for all verte-edge pairs ( i, i ) and ( i, i ), sch that ε W ( i, i )ε W ( i, i ) = σ( i ), ε W ( i, i ) ε W ( i, i ) = 0. () Every walk has eactly two opposite directions. A walk W with a direction ε W is called a directed walk, denoted (W, ε W ). 3

4 Lemma. The sign of a closed walk eqals ( ) k, where k is the nmber of times that the walk is incoherent at vertices, inclding the initial and terminal verte. Proof. We perform a short calclation that applies to open as well as closed walks. W = 0... n n be a walk, either open or closed. Then n n ( σ(w ) = σ( i ) = ε(i, i )ε( i, i ) ) i= i= n ( = ε( 0, )ε( n, n ) ε(j, j )ε( j, j ) ) j= { ( ) k ε(v 0, e )ε(v n, e n ) = ( ) k if W is open, if W is closed, where k is the nmber of times that W is incoherent at its internal vertices if W is an open walk. Let S E be an edge sbset. A reorientation by S is an orientation ε S obtained from ε by reversing the orientations of the edges in S and keeping the orientations of edges otside S nchanged. Ths ε S is given by { ε(v, e) if e S, ε S (v, e) = ε(v, e) if e / S. Sign-labeled covering graph. Let the end-vertices of each link edge of the graph G = (V, E) be ordered arbitrarily and fied. The sign-labeled covering graph of a signed graph Σ = (V, E, σ) is an ordinary graph Σ = (Ṽ, Ẽ) with the verte and edge sets Ṽ = V {±}, Ẽ = E {±}, defined as follows: If two vertices, v V are adjacent by an edge E, then the vertices (, α), (v, α σ()) in Ṽ are adjacent by an edge in Ẽ, and the vertices (, α), (v, α σ()) are adjacent by another edge in Ẽ. We denote by (, β) (with the inde β randomly selected) and by (, β). For simplicity, we write α = (, α), β = (, β), β = α v α σ(), β = α v α σ(). There is no canonical way to choose the inde β to label the edge between α and v α σ() with β, when the edge is a link between its two end-vertices, v. However, if we choose one of two orders between the end-vertices and v, say, v, we can choose labels as follows: α = α v α σ(), α =,. If so, we have = v σ() and = v σ(). Notice that the symbol β is jst a signlabeled edge in Σ, nothing to do with the sign σ() of the edge in Σ. All sign-labeled covering graphs are isomorphic, when varios orders are selected for the end-vertices of links in the nderlying signed graph. There is a natral graph homomorphism π : Σ Σ, called the projection from Σ to Σ, which is a pair of fnctions π V : Ṽ V and π E : Ẽ E, defined by π V ( α ) =, π E ( β ) =. 4 Let

5 We sally write π V and π E simply as π. There is a canonical involtory bt fied-point free graph atomorphism of Σ, called the agmentation of Σ, defined by ( α ) = α, ( β ) = β. When is a negative loop at its niqe end-verte, the edges, are two parallel edges in Σ with the end-vertices,. We may think of Ṽ as having a positive level V = { V } and a negative level V = { V }. We shall see later that it is impossible to lift all edges of Σ to the same levels when Σ is nbalanced. A positive edge is lifted to two edges staying inside each level, bt a negative edge is lifted to two edges crossing between the two levels. The orientation ε on Σ can be lifted to an orientation ε on Σ as follows, called the lift of ε. Let E be an edge incident with a verte V, and let be a lift of and be incident with a verte α Ṽ. Then we define ε( α, ) = α ε(, ). () Since α is an end-verte of the lifted edge, then by definition we have ε( α, ) = α ε(, ). Alternatively, if we have chosen an order between and v for each link = v, say, v, then we define ε( α, α ) = α ε(, ), ε(v α σ(), α ) = α σ() ε(v, ). (3) It is easy to see that the two arrows on each lifted edge are the same, regardless of the sign of the edge in Σ. In fact, for each edge with end-vertices, v (possibly = v), we have Analogosly, we have ε( α, ) ε(v α σ(), ) = α ε(, ) α σ() ε(v, ) =. ε( α, ) ε(v α σ(), ) = ( α ε(, ))( α σ() ε(v, )) =. This means that the two arrows on each of the two lifted edges, have the same directions; see Figre for a link edge and Figre for a loop edge. So ( Σ, ε) is an ordinary directed nsigned graph. Note the covering graph Σ is independent of the chosen order on the two v v v v v v v (a) σ() = v (b) σ() = v (c) σ() = Figre. Lifting of a link edge and its orientation end-vertices of each link edge in the sense that the covering graphs with respect to different orders are all graph isomorphic. Two oriented edges, y E incident with a common verte v, with the orientations ε(v, ) and ε(v, y), are said to be coherent at v if ε(v, ) ε(v, y) =, (4) 5

6 Figre. Lifting of a loop and its orientation which is eqivalent to ε(v, ) ε(v, y) = 0. We shall see that the lifting of orientations preserves coherence. If and y are oriented edges with a common verte v, with the orientations ε(v, ) and ε(v, y), then the concatenated lift of vy to v α ỹ, with the orientations ε(v α, ) and ε(v α, ỹ), is coherent at v. In fact, ε(v α, ) ε(v α, ỹ) = α ε(v, ) α ε(v, y) =. (5) Let ε i be orientations on signed sbgraphs Σ i, i =,. The copling of ε and ε is a fnction [ε, ε ] : E {, 0, }, defined (for each edge with its end-verte ) by if Σ Σ, ε (, ) = ε (, ), [ε, ε ]() = if Σ Σ, ε (, ) ε (, ), (6) 0 otherwise. One may etend ε i to Σ by reqiring ε(v, y) = 0 whenever the edge y in not incident with the verte v in Σ i. We always assme this etension atomatically. Then alternatively, [ε, ε ]() = ε (, ) ε (, ), where End(). The lifted graphs Σ i are sbgraphs of Σ and ( Σ i, ε i ) are oriented sbgraphs of the oriented graph ( Σ, ε). Moreover, the lifting of orientations preserves the copling, that is, for each lift β of an edge. Indeed, [ ε, ε ]( β ) = [ε, ε ]() (7) [ ε, ε ]( β ) = ε ( α, β ) ε ( α, β ) = α ε (, ) α ε (, ) = [ε, ε ](). Let W = 0... n n be a walk in Σ of length n with a direction ε W. We may lift W to a walk W (called a lift of W ) in Σ as follows: Select an initial verte α 0 ; define W = α 0 0 α... α n n n αn n, α i = α i σ( i ), i = α i i α i i. (8) A lift W is call a resoltion of W if it is a cycle or an open path. There are eactly two lifts of W in the form (8), since there are eactly two choices for α 0. Moreover, it follows from (5) that the lifted orientation ε W from ε W by () to the edges in W forms a direction of W. Ths (W, ε W ) is lifted eactly to two directed walks ( W, ε W ). We call W the projected walk of W and write W = π( W ). 6

7 Lemma. Let W be a closed walk in Σ with initial and terminal vertices, and let ε W be a direction of W with respect to the initial verte. Let ( W, ε W ) be a lift of the directed walk (W, ε W ). If W is positive, then ( W, ε W ) is a directed closed walk. If W is negative, then ( W, ε W ) is a directed open walk. Proof. Let W = 0... n n n and be lifted to a walk W = α 0 0 α... n αn n. Since positive edges are lifted to the edges staying in the same level and negative edges to the edges crossing the two levels, we see that σ( i ) = α i α i. Then n n σ(w ) = σ( i ) = α i α i = α 0 α n. i= Clearly, σ(w ) = if and only if α n = α 0, and sbseqently if and only if W is closed. i=. Flows Flows on signed graphs. The incidence matri of an oriented signed graph (Σ, ε) is the matri M = M(Σ, ε) = [m(, )] indeed by the set V E, where m is a fnction m : V E Z defined by ε(, ) if is a link, m(, ) = ε(, ) if is a negative loop, 0 otherwise. Alternatively, m(, ) = v End(), v= ε(v, ), where End() is a mltiset of two identical vertices when is a loop. An integral flow on an oriented signed graph (Σ, ε) is a fnction f : E Z which is conservative at every verte, meaning that f = 0, where : Z E Z V is called the bondary operator of (Σ, ε), defined by f() = m(, )f() = ε(, )f(). (9) E E v End(), v= The vale of the fnction f at is also called the ecess of f at in network flows. The set of all integral flows on (Σ, ε) forms a Z-modle, called the flow lattice by Chen and Wang, who developed its basic theory in [5]. One can define flows with vales in an arbitrary abelian grop, sch as the additive reals and finitely generated abelian grops. Many of the following remarks are applicable in general, so we omit the word integral. The theory of flows of signed graphs depends essentially on the graph and sign fnction bt not on the orientation, since for two orientations ε, ρ there is an isomorphism from the flow lattice of (Σ, ε) to the flow lattice of (Σ, ρ) by f [ε, ρ]f. The spport of a fnction f : E Z is the set of edges sch that f() 0, denoted spp f. We denote by Σ(f) the signed sbgraph of Σ whose edge set is spp f and verte set consists of vertices incident with some edges in spp f. The flow that is zero on all edges is called the zero flow. Flows other than the zero flow are referred to nontrivial flows. A circit flow is a flow whose spport is a signed-graph circit, having vales ± on the edges of the cycles and vale ± on the edges of the circit path (for Type III circits). See Figre 3 for signed-graph circit flows of Type II and Type III. The theory of flows on ordinary graphs is simply the case that all edges are positive, where a circit flow has a cycle as its spport and has vales ± on the edges of the cycle (see [], 7

8 (a) Type II (b) Type III Figre 3. Signed-graph circit flows of Types II and III p.5). Signed-graph circit flows are defined analogosly in [5]. Nowhere-zero integral flows on signed graphs are stdied in [, 4, ]. We say that an integral flow f conforms to the sign pattern of f if spp f spp f, and f () has the same sign as f() for all edges in spp f. An integral flow f on (Σ, ε) lifts to a flow on the oriented sign-labeled covering graph ( Σ, ε), possibly in more than one way. The best way to see this is throgh the correspondence between flows and walks, which eists if Σ(f) is connected. A directed closed positive walk (W, ε W ) on (Σ, ε) corresponds to a niqe integral flow f (W, εw ), defined by f (W, εw )() = [ε, ε W ](y), (0) y W, y= where W is viewed as a mltiset {,,..., n } of edges if W = 0... n n ; see [5]. Clearly, f (W,εW )() is the nmber of times that the walk W traverses the edge along the direction ε mins the nmber of times that W traverses along the direction opposite to ε. Whenever the direction ε W is the same as ε on W, we simply write f (W, εw ) as f W, and f W () = #{y W y = } (as a mltiset). () To see that f (W, εw ) is a flow, consider the contribtion to f (W, εw ) of a pair of consective edges, i i i, at the intervening verte i. Since (W, ε W ) is coherent at i, the contribtion of these edges to f (W, εw )( i ) is 0. This same argment applies to the initial verte if we take sbscripts modlo the length of W. We can apply the same definition f (W, εw ) to any directed walk (W, ε W ) (not necessarily closed), however the reslt is no longer a flow. In fact, we have the following lemma. Lemma 3. Let (W, ε W ) be a directed walk with W = 0... n n n. fnction f (W, εw ) is conservative everywhere ecept at 0 and n, that is, Then the f (W, εw )() = 0, 0, n. Moreover, if W is closed, then { 0 if W is positive, f (W, εw )( 0 ) = ε W ( 0, ) if W is negative. If W is open, then f (W, εw )( 0 ) = ε W ( 0, ) and f (W, εw )( n ) = ε W ( n, n ) = σ(w ) ε W ( 0, ). 8

9 Proof. Fi a verte. Let be appeared in the verte-edge seqence of W as the sbseqence l, l,..., lk. If 0, n, then ε W ( li, li ) = ε W ( li, li ). We compte f (W, εw )() = m(, ) [ε, ε W ](y) E y W, y= = y W m(, y)[ε, ε W ](y) = y W = y W v End(y), v= v End(y), v= ε(v, y)[ε, ε W ](y) ε W (v, y). The last eqality follows from definition of the copling. For each verte li with i k, we have the sb-walk segment li li li, which contribtes eactly the two terms ε W ( li, li ), ε W ( li, li ) in the above last sm. Ths f (W, εw )() = k [ε (W, εw )( li, li ) ε W ( li, li )] = 0 i= regardless of whether W is closed or open. Assme W is closed, that is, 0 = n. If = 0, then l = 0 and l k = n. So l and lk are the initial and terminal vertices of W, 0 = n and n =. We have f (W, εw )( 0 ) = ε W ( 0, ) ε( n, n ). In particlar, if W is positive, we have f (W, εw )( 0 ) = 0; if W is negative, we have f (W, εw )( 0 ) = ε( 0, ) = ε W ( n, n ). Assme W is open, that is, 0 n. If = 0 or = n, then f (W, εw )( 0 ) = ε W ( 0, ) and f (W, εw )( n ) = ε W ( n, n ) = σ(w )ε W ( 0, ). Conversely, directed closed walks (which are sally not niqe) can be constrcted from integral flows. Chen and Wang [6] developed a flow redction algorithm to obtain an eqivalent classification of indecomposable flows. The method employed in [6] is an algorithmic approach. The present paper is a strctral approach by resoltion of singlarities. Proposition 4. Let f be a nonnegative, nontrivial, integral flow on (Σ, ε) sch that Σ(f) is connected. Then there eists a directed, closed, positive walk (W, ε) sch that f W = f. Proof. We apply indction on the total weight f := E f(). Choose a verte 0 and an edge incident with 0 in Σ(f). Let be the other end-verte of ( = 0 if is a loop). This gives a walk W = 0 of length. Clearly, f f W 0. Now assme that we have selected a partial walk W k = 0... k k and f f Wk 0. If W k is not a closed positive walk, that is, W is open, or closed bt is negative, then by Lemma 3, the fnction f Wk is not a flow, for it is not conservative at k. Since f is conservative at k, the fnction f f Wk cannot be conservative at k. Then there eists 9

10 an edge k incident with k in Σ(f f Wk ) sch that ε( k, k ) = ε( k, k ). Let k denote the other end-verte of k and etend W k to W k = W k k k. Clearly, we have f f Wk 0 by the constrction. Contine this procedre when W k is not a closed positive walk. We finally obtain a directed closed positive walk (W n, ε). Ths f (= f f Wn ) is a nonnegative integral flow and clearly f < f. Let Σ(f ) be decomposed into connected components Σ i, and set f i = f Σi. Then f = i f i and spp f i = E(Σ i ). It is easy to see that f i are nonnegative nontrivial integral flows of (Σ, ε) and f i < f. By indction, there eist directed closed positive walks (W i, ε) sch that f i = f W i. It is clear that the nion of all (W i, ε) and (W n, ε) is connected. One can constrct a single directed closed positive walk (W, ε) by rearranging the initial and terminal vertices of all W i and W n, and connecting them properly at some of their intersections. Clearly, we have f W = f by the constrction. Let f be an integral flow of (Σ, ε). Associated with f is an orientation ε f on Σ defined by (for each edge at its end-verte ) { ε(, ) if f() 0, ε f (, ) = () ε(, ) if f() < 0. Corollary 5. Let f be a nontrivial integral flow on (Σ, ε). If Σ(f) is connected, then there eists a closed positive walk W sch that (W, ε f ) is a directed closed walk and f (W, εf ) = f. Proof. The absolte fnction f = [ε, ε f ]f is a nonnegative, nontrivial, integral flow of (Σ, ε f ). According to Proposition 4, there eists a directed closed positive walk (W, ε f ) sch that f W = f within the oriented signed graph (Σ, ε f ), where f W () = [ε f, ε f ](y) =, E. y W, y= y W, y= For the same directed closed positive walk (W, ε f ) within (Σ, ε), we have f (W, εf )() = [ε, ε f ](y) = [ε, ε f ]() y W, y= = [ε, ε f ]()f W (), E. Since f = [ε, ε f ] f and f = f W, it follows that f (W, εf ) = f. y W, y= Lifted flows. Consider a fnction f : Ẽ Z defined on the edge set of the sign-labeled covering graph Σ. The projection of f is the fnction π( f) : E Z defined by π( f)() = f( ) f( ). (3) Let M = M( Σ, ε) denote the incidence matri of ( Σ, ε) of size Ṽ Ẽ. Write M = [ m( α, )]. Then m( α, ) = ε( α, ) if is a link edge with end-verte α, and m( α, ) = 0 otherwise. Since only positive loops of Σ are lifted to loops in Σ, we have { 0 if is a positive loop, m( α, ) = (4) α ε(, ) otherwise. 0

11 The bondary operator : ZẼ ZṼ is defined by f( α ) = m( α, y) f(y). y Ẽ Lemma 6. Let f be a fnction defined on Ẽ. Then π( f) is given by If f is a flow on ( Σ, ε), so is π( f) on (Σ, ε). π( f)() = f( ) f( ), V. (5) Proof. Fi a verte in Σ. Let E denote the set of all edges incident with. Then E is partitioned into the disjoint sets Lk(), Lp() of links and loops at respectively, and Lp() is frther partitioned into the disjoint sbsets Lp () = {positive loops at }, Lp () = {negative loops at }. Since m(, ) = 0 if is a positive loop at, we compte π( f)() = E m(, ) [ f( ) f( ) ] = Lk() Lp () ε(, )[ f( ) f( )] (6) ε(, )[ f( ) f( )]. (7) Fi a verte α of Σ. The link edges incident with α are the edges α with Lk() and the edges α, α with Lp (). We compte f( α ) = m( α, α ) f( α ) m( α, α ) f( α ) = α Lk() Lp () Lk() Lp () Lk() Lp () ε(, ) f( α ) α Lp () Lp () Lp () ε(, ) f( α ). The second eqality follows from (4). In particlar, we have f( ) = ε(, ) f( ) ε(, ) f( ); f( ) = Lk() Lp () ε(, ) f( ) Lp () ε(, ) f( ). Adding f( ) and f( ) together and comparing the reslt with π( f)() in (6), we obtain (5). Whenever f is a flow of ( Σ, ε), then f( ) = f( ) = 0. Ths π( f)() = 0. This means that π( f) is a flow of (Σ, ε). A lift of an integral flow f of (Σ, ε) to Σ is an integral flow f of ( Σ, ε) sch that π( f) = f. Proposition 7. (a) Let ( W, ε W ) be a lift of a directed closed positive walk (W, ε W ). Then π[f ( W, εw ) ] = f (W, ε W ). (8)

12 (b) Let f be an integral flow on (Σ, ε), and let (W, ε f ) be a directed closed positive walk sch that f = f (W, εf ). Let (W, ε f ) be lifted to a directed closed walk ( W, ε f ) in Σ. Then f := f ( W, εf ) is a flow on ( Σ, ε) lifted from f. Moreover, if f is nonnegative, so is f. Proof. (a) It sffices to show that f (W, εw )() = f ( W, εw ) ( ) f ( W, εw ) ( ). Since the lifting of orientations preserves the copling, then by (7) we have π[f ( W, εw ) ]() = f ( W, ε W ) ( ) f ( W, εw ) ( ) = [ ε, ε W ](ỹ) ỹ W, ỹ= or = [ε, ε W ](y) = f (W, εw )(). y W, y= (b) Since f = f (W, εf ), it is trivial by (8) that f is a lifted flow of f. If f is nonnegative, then ε f = ε; sbseqently, ε f = ε. Hence f = f ( W, εf ) is nonnegative by definition (0). An integral flow f on an oriented signed graph (Σ, ε) is said to be conformally decomposable if it is nontrivial and can be represented as a sm of two other integral flows, f = f f, each of which is nontrivial and conforms to the sign pattern of f, that is, f ()f () 0 for all edges ; this means that both f (), f () are positive or both are negative if they are nonzero. An integral flow of (Σ, ε) is said to be conformally indecomposable if it is not conformally decomposable. A nonnegative nontrivial integral flow f on (Σ, ε) is said to be minimal provided that if g is a nonnegative nontrivial integral flow on (Σ, ε) sch that g() f() for all edges, then g = f. If an integral flow f is nonnegative, then minimality is eqivalent to conformal indecomposability. In fact, if f is decomposed into f = f f, then f, f mst be nonnegative nontrivial integral flows and f f, f f; this means that f is not minimal. Conversely, if f is not minimal, say, there is a nonnegative nontrivial integral flow g on (Σ, ε) sch that g f bt g f, then h = f g is nontrivial and nonnegative, and f is decomposed into f = g h. In the following we shall see that the conformal indecomposability of a nontrivial integral flow f on (Σ, ε) is eqivalent to the minimality of the absolte vale flow f on (Σ, ε f ), where ε f is the orientation given by (). It is well known and easy to see that conformally indecomposable flows on an ordinary graph are jst graph circit flows. Lemma 8. A nontrivial integral flow f of (Σ, ε) is conformally indecomposable if and only if the absolte vale fnction f is a minimal flow of (Σ, ε f ). Proof. Applying the bondary operator (9), it is clear that f is a flow on (Σ, ε) if and only if f = [ε, ε f ] f is a flow on (Σ, ε f ). Since f is nonnegative, its minimality is eqivalent to its conformal indecomposability. We show necessity first. Sppose f is not minimal, that is, f = g g, where g i are nonnegative nontrivial integral flows of (Σ, ε f ). Then f i = [ε, ε f ] g i are nontrivial integral flows of (Σ, ε). Ths f = [ε, ε f ] f = [ε, ε f ] g [ε, ε f ] g = f f, and f f = g g 0, meaning that f is conformally decomposable. This is a contradiction. For sfficiency, sppose f is conformally decomposable, that is, f = f f, where f i are nontrivial integral flows of (Σ, ε) sch that f f 0. Then g i = [ε, ε f ] f i are nontrivial

13 integral flows of (Σ, ε f ). For each edge, if f i () > 0, we mst have f() > 0 and [ε, ε f ]() = ; if f i () < 0, we mst have f() < 0 and [ε, ε f ]() = ; then g i () 0. Hence f = [ε, ε f ] f = [ε, ε f ] f [ε, ε f ] f = g g, meaning that f is conformally decomposable. This is a contradiction. 3. Indecomposable flows A signed graph Ω with nonempty edge set is said to be Elerian if there eists a directed closed walk (W, ε W ) that ses every edge of Ω at least once bt at most twice, and the direction ε W has the same orientation on each pair of repeated edges. An Elerian signed graph is frther said to be prime if its directed closed walk does not properly contain directed closed sbwalks. An Elerian signed graph is said to be mimimal if it does not properly contain Elerian signed sbgraphs. It is easy to see that minimal Elerian signed graphs mst be prime Elerian signed graphs. Definition 9. A signed graph T with nonempty edge set is called a cycle-tree if it satisfies the following conditions: (a) T is connected. (b) Each block of T is either a cycle (called block cycle) or an edge. (c) Each ct-verte is incident with eactly two blocks. (d) Each block incident with eactly one ct-verte is a cycle (called an end-block cycle). A cycle-tree is said to be Elerian if it frther satisfies (e) Parity Condition: The sign of a block cycle eqals ( ) p, where p is the nmber of ct-vertices of T on the cycle. We shall see that prime Elerian signed graphs are Elerian cycle-trees, and minimal Elerian signed graphs are signed-graph circits, i.e., positive cycles or handcffs. A cycle-tree is indeed a tree-like signed graph whose vertices are the block cycles and edges are the paths (of possible zero length, called block paths) between pairs of block cycles. The end-vertices of block paths are ct-vertices. If a block path has length zero, it is a common ct-verte of two block cycles. We may also think of an Elerian cycle-tree as a tree whose vertices are some verte-disjoint maimal Elerian sbgraphs and edges are the paths (of positive length) between the Elerian sbgraphs, where each sch maimal Elerian graph is also a tree-like whose vertices are edge-disjoint cycles and edges are the intersection vertices between pairs of cycles. Figre 4 ehibits an Elerian cycle-tree with a direction. Figre 4. An Elerian cycle-tree with a direction Let T be an Elerian cycle-tree. A direction of T is an orientation ε T on T sch that (T, ε T ) has neither sorces nor sinks, bt for each block cycle C the signed sbgraph (C, ε T ) 3

14 has either a sorce or a sink at each ct-verte of T on C. It is easy to see that there eist eactly two (opposite) directions on T. For nsigned graphs, since all edges are positive, Elerian cycle-trees are jst cycles, and directed Elerian cycle-trees are directed cycles. When T is contained in a signed graph Σ, the indicator fnction of T is the fnction I T : E Z defined by I T () = if belongs to a block cycle, if belongs to a block path, 0 otherwise. The fnction [ε, ε T ] I T is called the characteristic vector of (T, ε T ) within (Σ, ε). A closed walk on T is called an Elerian tor if it ses every edge of T and has minimm length. A sbgraph of T is called an Elerian cycle-sbtree if it is also an Elerian cycle-tree and its block cycles and block paths are block cycles and block paths of T. Proposition 0 (Eistence and Uniqeness of Direction on Elerian Cycle-Tree). Let T be an Elerian cycle-tree. Then (a) There eists a closed walk W on T sch that (i) W ses each edge of block cycles once and each edge of block paths twice, (ii) whenever W meets a ct-verte, it crosses from one block to another block. Moreover, each sch closed walk is an Elerian tor on T, having the length l(t ) := i l(c i ) j (9) l(p j ), (0) where C i are block cycles and P j are block paths of T. (b) Each Elerian tor W on T satisfies the conditions (i), (ii), and l(w ) = l(t ). (c) There eists a niqe direction ε T of T (p to opposite sign) sch that (W, ε T ) is coherent for each Elerian tor W on T and f (W, εt ) = [ε, ε T ] I T. () Moreover, if is a ct-verte and W = W W, where W i are closed sb-walks having initial and terminal vertices at, then both W i are negative and (W i, ε T ) are incoherent at and coherent elsewhere. Proof. (a) & (b): Let T be the sign-labeled covering graph of T. We claim that there eists a directed cycle ( W, ε W ) sch that (i ) W covers each edge of block cycles once and each edge of block paths twice, (ii ) orientations on each edge of T indced from ( W, ε W ) by the projection are identical, and (iii ) the indced orientation ε T from ε W is a direction of T. We proceed by indction on the nmber of block cycles. If there is only one block cycle, the cycle mst be positive as there is no ct vertices. Choose a direction of the cycle. The directed cycle lifts to a directed cycle in T by Lemma. Now we assme that T contains at least two block cycles. Choose an end-block cycle C 0 of T with the niqe ct-verte 0. There eists a path P connecting C 0 to another block cycle C, having the initial verte 0 on C 0 and the terminal verte v 0 on C. Let C 0, P be written as C 0 = 0... m m m, P = v 0 y v y... v n y n v n, where 0 = m = v n, and be lifted to T as the paths P 0 = α 0 0 α... α m m m αm m, P = v β 0 0 ỹ v β ỹ... v β n n ỹ n v βn 4 n.

15 Note that α m = α 0, since the cycle C 0 is negative. Let s remove C 0, P from T, change the sign of an edge z in C at v 0, and rename z as z. We then obtain an Elerian cycle-tree T, which has one fewer block cycles than T. By indction there eists a directed cycle ( W, ε W ) satisfying the conditions (i ) (iii ). Let z be an edge of W that covers z, having an end-verte v γ 0 covering v 0. Let s write W as a closed path from v γ 0 to v γ 0, having arranged z as the last edge. Let s = ε W (v γ 0, z ), and let ( P, ε P ) be a directed path in T from v γ 0 to v γ 0, obtained from ( W, ε W ) by replacing the edge z with an edge z of T at v γ 0, where z covers z, having the orientation ε P (v γ 0, z) = s. In the case that P has length zero, we have 0 = v 0. Let α 0 = γ. Then P 0 is an open path from γ 0 to γ 0. Choose a direction ε P0 of P 0 sch that ε P0 ( α 0 0, ) = s. Then ( W, ε W ) is a directed cycle in T, satisfying the conditions (i ) (iii ), where W = P 0 P and ε W = ε P0 ε P. In the case that P has positive length, let β 0 = γ. Then β n is determined by P. Let α 0 = β n. Then P P 0 is an open path from v γ 0 to α 0 0. Choose a direction ε P of P of P0 sch that m, m ) = s. Consider the path sch that ε P (v β 0 0, ỹ ) = s, then ε P (v βn n, ỹ n ) = s. Choose a direction ε P0 ε P0 ( α 0 0, ) = s, then ε P0 ( αm P = v β 0 0 ỹ v β ỹ... v β n n ỹ nv βn n ; its reverse P is a path from αm 0 to v γ 0. There is an indced direction ε P on P by the agmentation, that is, ε P (v β 0 0, ỹ) = s, ε P (v βn n, ỹn) = s. Note that the direction ε P of P indces a direction ε P on P by the projection, that is, ε P (v 0, y ) = β 0 s, ε P (v 0, y n ) = β n s. The direction ε P of P also indces the direction ε P by the projection. Ths ( W, ε W ) is a directed cycle in T, satisfying the conditions (i ) (iii ), where W = P P 0P P and ε W = ε P ε P0 ε P ε P. We define the closed walk W = π( W ) and its direction ε W = π(ε W ). Then W satisfies the conditions (i), (ii), and is an Elerian tor on T. Conversely, each Elerian tor W on T mst pass each edge of block paths at least twice, for each edge of block paths is a ct-edge. So W has length at least l(t ). Since W has minimm length, it follows that l(w ) = l(t ). The minimm property of l(w ) forces that W satisfies the conditions (i) and (ii). (c) Let ε T be the direction of T by taking the projection of the direction ε W. Now for an arbitrary Elerian tor W on T, the direction property of ε T forces (W, ε) to be a directed closed walk. The niqeness of ε T p to opposite sign follows from the tree strctre of T. The properties (i) and (ii) imply that I T is a flow of (T, ε T ). Ths f (W, εt ) = [ε, ε T ] I T. It is easy to see that the nmber of Elerian tors (p to the reversing of orders of verteedge seqences) on an Elerian cycle-tree T is q, where q is the nmber of ct-vertices of T. Lemma (Minimality of Elerian Cycle-Tree). Let T be an Elerian cycle-tree. If a signed sbgraph T of T is also an Elerian cycle-tree, then T = T. Proof. The block cycles of T are certainly block cycles of T. Sppose T is properly contained in T. Then there eist a block cycle C of T and a verte of C sch that is not a ctverte in T bt a ct-verte in T. Let ε T be a direction of T and ε T a direction of T sch that they agree on C. Then Proposition 0(c) implies that ε T T = ε T. Note that (C, ε T ) 5

16 mst be coherent at when C is considered as a block cycle in T, and mst have either a sink or a sorce at when C is considered as a block cycle in T. This is a contradiction. Lemma (Resoltion of Indecomposable Flows). Let f be a conformally indecomposable flow on (Σ, ε). Then f can be lifted to a conformally indecomposable flow f on ( Σ, ε). So Σ( f) is a cycle. Proof. It is clear that the conformal indecomposability of f implies that Σ(f) is connected. Let f = f (W, εf ), where (W, ε f ) is a directed closed positive walk of Σ. Lift (W, ε f ) to a directed closed walk ( W, ε f ) in Σ. Then f is lifted to a flow f ( W, εf ) of ( Σ, ε) by Proposition 7(a), denoted f = f ( W, εf ). Sppose f is decomposed into f = f f, where f i are nontrivial flows and f f 0. Notice that f = [ε, ε f ] f, f = [ ε, εf ] f, fi = [ ε, ε f ] f i. Denote f i = π( f i ), which are nontrivial flows of (Σ, ε). Then for each edge of Σ, f i () = [ ε, ε f ]( ) f i ( ) [ ε, ε f ]( ) f i ( ) = [ε, ε f ]() ( f i ( ) f i ( ) ) = [ε, ε f ]() π( f i )(). Taking absolte vales of both sides, we obtain f i = π( f i ); sbseqently, f i = [ε, ε f ] f i. Ths f = f f and f f = f f 0, meaning that f is conformally decomposable. This is a contradiction. Theorem 3 (Classification of Indecomposable Flows). Let f be a flow of (Σ, ε) and Ω = Σ(f). Then f is conformally indecomposable if and only if Ω is an Elerian cycle-tree with a direction ε Ω = ε f Ω and f = [ε, ε Ω ] I Ω. () Proof. We show necessity first. Recall that ε f is an orientation obtained from ε by reversing the orientations on the edges sch that f() < 0. Then f = [ε, ε f ] f is a conformally indecomposable flow on (Σ, ε f ); so is a minimal flow becase of nonnegativity. Let ε f be lifted to an orientation ε f on Σ and set ε Ω = ε f Ω. Let W = 0... n n be a closed positive walk sch that (W, ε Ω ) is directed. Then f = f W within (Σ, ε f ) by Corollary 5 and Lemma 8. Let W be lifted to a walk W = α 0 0 α... n αn n in Σ. Then ( W, ε Ω ) is a directed closed walk in Σ by Lemma, and f W is a conformally indecomposable flow of ( Σ, ε f ) by Lemma. Since Σ is an nsigned graph, the closed walk W mst be a cycle. Since Σ is a doble covering of Σ and π( W ) = W, it follows that W has only possible doble vertices and doble edges. If W has no doble vertices, that is, W has no self-intersections, then (W, ε Ω ) is a directed cycle. Clearly, its nderlying signed graph is a balanced cycle, of corse is an Elerian cycle-tree contained in Σ, and f = [ε, ε Ω ] I Ω. Assme that W has some self-intersections. Let be a doble verte of W. Rewrite W = W W, where W i are closed walks with the initial and terminal vertices at ; more specifically, W = 0... m m, W = m m m... n n with 0 = m = n =. We claim that W i are negative, (W i, ε Ω ) are incoherent at and coherent elsewhere, and is a ct-verte. 6

17 Notice that (W i, ε Ω ) are coherent everywhere ecept at. Sppose (W, ε Ω ) is coherent at. Then (W, ε Ω ) is also coherent at. Ths (W i, ε Ω ) are directed closed walks. We then have f W = f W f W within (Σ, ε f ), meaning that f is conformally decomposable; this is a contradiction. Hence (W i, ε Ω ) mst be incoherent at. Lemma 3 implies that the closed walks W i are negative. Let s write W = W W, where W = α 0 0 α... m αm m, W = αm m m α m m... n αn n are open simple paths. Then α 0 = α n = α m by Lemma. Sppose is not a ct-verte, that is, the walks W and W meet at a verte v other than. Write the verte v in W, W as k, h respectively, that is, v = k = h, where k m and m h n. Since W is a cycle, then α k k α h h ; sbseqently, α k = α h. Consider the closed walk W = α 0 0 α... α k k k α k k ( α h h ) h α h h... α m m m αm m ( α 0 0 ). Let s = ε Ω ( α 0 0, ). Then ε Ω ( α k 0, k ) = s, for the open walk ( W, ε Ω ) is directed. Analogosly, s = ε Ω ( αm m, m ) = α m ε Ω ( m, m ), the second eqality is by definition () of lifting orientation. Then ε Ω ( αm m, m) = α m ε Ω ( m, m ) = s, that is, ( W, ε Ω ) is coherent at α 0 0. Ths ( W, ε Ω ) is a directed closed walk in Ω. It follows that (W, ε Ω ) is a directed closed walk in Ω, and is contained in the walk W as edge mltisets. Therefore, f W f W. The minimality of f W implies that f W = f W ; this is impossible, for W is properly contained in W as mltisets. Now the closed walk W has only possible doble vertices and doble edges, and all doble vertices are ct-vertices. The signed sbgraph Ω is obtained from the cycle W by the projection, identifying some pairs of vertices and some pairs of edges. The nidentified vertices and edges form the block cycles C i, and the connected components of identified vertices and edges form the block paths P j (of possible zero length) between some pairs of the cycles C i. More specifically, viewing each connected component of identified edges and vertices as a single path-identification, then each path-identification transforms a cycle eactly into two cycles, the identified path (of possible zero length) connects the two cycles and becomes a block path. Since each doble verte of W is a ct-verte of Ω, it trns ot that Ω is a tree-like with vertices C i and edges P j ; the cycles C i mst be blocks, the vertices and edges of P j are ct-vertices and ct-edges respectively. Recall the incoherence of (W i, ε Ω ) at the doble verte. It follows that (C i, ε Ω ) are incoherent at the ct-vertices of Ω on C i and coherent elsewhere. Ths the cycles C i has sign ( ) p by Lemma, where p is the nmber of ct-vertices on C i. We have finished proof that Ω is an Elerian cycle-tree, ε Ω is a direction of Ω, and (W, ε Ω ) is a directed Elerian tor. Since each edge on the block cycles appears once in W and each edge on the block paths appears twice in W, we see that f W = I Ω within (Σ, ε f ). Therefore f = f (W, εω ) = [ε, ε Ω ] I Ω. Conversely for sfficiency, let s write [ε, ε Ω ] I Ω = k i= f i, where f i are conformally indecomposable flows of (Σ, ε). Let (Ω i, ε Ωi ) be Elerian cycle-trees sch that f i = [ε, ε Ωi ] I Ωi. Since each f i conforms to the sign pattern of f, then Ω i is a signed sbgraph of Ω and ε Ωi is the restriction of ε Ω on Ω i. The block cycles of Ω i are certainly block cycles of Ω. Consider a possible block path P of Ω i. If l(p ) = 0, then P is the intersection of two block cycles C, C of Ω i. Clearly, the block cycles C, C of Ω i mst be block cycles of Ω, and P is certainly contained in Ω. If l(p ) > 0, then f i () = ± for all edges of P. Since f i conforms to the sign pattern of f, it forces that [ε, ε Ω ] I Ω () = ± for all edges of P. This means that P 7

18 is a block path of Ω. This means that Ω i is an Elerian cycle-tree of Ω. Ths Ω = Ω i by Lemma. Therefore k = and f = [ε, ε Ω ] I Ω is conformally indecomposable. Proposition 4. A signed graph Ω is prime Elerian if and only if Ω is an Elerian cycletree. Proof. If Ω is an Elerian cycle-tree, then by Proposition 0 there eist an orientation ε Ω on Ω and a closed walk that ses every edge of Ω once bt at most twice sch that (W, ε Ω ) is a directed closed positive walk. It is clear from the strctre of Elerian cycle-tree that (W, ε Ω ) does not contain properly directed closed positive sbwalks. So Ω is a prime Elerian signed graph. Conversely, if Ω is a prime Elerian signed graph, then by definition there eist an orientation ε Ω on Ω and a closed walk W that ses every edge of Ω at least once bt at most twice, sch that (W, ε Ω ) is a directed closed positive walk and does not properly contain directed closed positive sbwalks. Since f W is a flow of (Ω, ε Ω ), there is a conformally indecomposable flow f sch that f W f 0. By Theorem 3 there eist an Elerian cycle-tree T and its direction ε T sch that f = [ε Ω, ε T ]I T. Then we mst have ε T = ε Ω on T. Let (W, ε T ) be a directed closed positive walk on T that follows the direction of W. Then (W, ε T ) is a directed closed positive sbwalk of (W, ε Ω ). The primeness of (W, ε Ω ) implies that (W, ε Ω ) = (W, ε T ). Hence Ω = T. Theorem 5 (Half-integer Scale Decomposition). Let Ω be an Elerian cycle-tree with a direction ε Ω. If Ω is not a signed-graph circit, then there eists a closed positive walk W = C 0 P C P... P n C n P n, n on Ω, satisfying the for conditions. (a) C i are all end-block cycles, and P i are paths of positive lengths. (b) Each edge of non-end block cycles appears in eactly one of P i, and each edge of block paths appears in eactly two of P i. (c) Each (C i P i C i, ε Ω ) is a directed circit of Type III with C n = C 0. (d) I Ω = n i=0 I Σ(C i P i C i ). Proof. Let C 0, C,..., C n be end-block cycles with niqe ct-vertices 0,,..., n respectively. Let Q j be block paths, and R k the paths on non-end block cycles between two net ct-vertices. Let C i be lifted to two disjoint directed open paths C i, C i between i and i. Let Q j be lifted to two disjoint directed open paths Q j, Q j. And let R k be lifted to disjoint directed open paths R k, R k. Each C i α connects eactly two paths Q j, Q j and Q C j i α Q j is a directed open path. Likewise, each R γ k connects eactly two paths Q β β j, Q j with j j and Q β R γ β j k Q j is a directed open path; sbseqently, Q β j R γ k Q β j is a directed open path. Let s partition the collection { R k, R k } of open paths into two disjoint sb-collections { R γ k k } and { R γ k k } arbitrarily. Then the nion of the paths in { Q j, Q j, R γ k k } is a collection of disjoint directed open paths, having initial and terminal vertices α i, α i with i i. Let P denote that the directed open path with from a verte β 0 0 and its terminal verte γ let P denote the open path from β s (β = γ ) to γ from β s (β = γ ) to γ 3 paths s ; and let P 3 denote the open path s 3. Contine this procedre, we obtain a seqence of directed open P, P,..., Pm, 8 s ;

19 where P i is from β i s i to γ i s i with β i = γ i, and γ m s m = γ 0 0 with β 0 = γ 0. Note that both P i C si Pi and P i C si Pi are directed open paths for all i from 0 to n. Withot loss of generality, let s denote by P the open path from s 0 0 to t Let s partition the collection { C i, C i { C α i i } and { C α i i } of open paths into two disjoint sb-collections } arbitrarily. It follows that W = C α 0 0 P Cα P... P n Cα n n P n is a directed closed walk in ( Σ, ε). Note that the projections of C α i i are nbalanced cycles C i in Σ, and (C i, ε) is coherent everywhere ecept incoherence at i. Let P i be the projection of P i. Then the projection of W is the directed closed walk W = C 0 P C P... C n P n in (Σ, ε), and each (C i P i C i, ε) is a directed circit of Type III with C n = C 0. Clearly, each edge of non-end-block cycles appears in eactly one of the paths P i, and each edge of block paths appears in eactly two of the paths P i. Hence I Ω = f W and I Ω = I C0 I Pn n i= ( ICi I Pi ) = n I Σ(Ci P i C i ). The weights on the edges of the cycle-tree in Figre 4 form a conformally indecomposable flow with respect to the given direction there, which is eactly the characteristic vector of the cycle-tree. This conformally indecomposable flow can be decomposed into one-half of the sm of three signed-graph circit flows as demonstrated in Figre 5. i=0, C C 0 C C C = C C 0 C0 C Figre 5. A conformally indecomposable flow is decomposed conformally into one-half of signed-graph circit flows of Type III. The half-integer phenomenon in Theorem 5(d) is also appeared in a reslt of Geelen and Genin [9] (Corollary.4, p. 83), thogh or problem of classifying conformally indecomposable flows and certain optimal problem considered by Geelen and Genin for signed graphs are completely different. The half-integer phenomenon in both cases is a conseqence of sign on the edges of signed graphs. One may consider decomposition of integral flows withot conforming the sign patterns. It is clear that every nontrivial integral flow is a positive integral linear combination of conformally indecomposable flows. We shall see that each conformally indecomposable flow can be frther decomposed into integral linear combination of signed-graph circit flows 9

20 C4 Figre 6. A maimal independent set of the Elerian cycle-tree in Fig. 4 withot confirming the sign patterns. So each integral flow is an integral linear combination of signed-graph circit flows. This fact is already eplicitly given in terms of a maimal independent set (=matroid basis) in [5] (see Eq. (4.7) of Theorem 4.9, p. 75). For instance, the signed graph in Figre 6 is an maimal independent set of the Elerian cycle-tree in Figre 4. The conformally indecomposable flow in Figre 4 is frther decomposed into signed-graph circit flows in Figre 7 withot confirming the sign patterns. We smmarize the fact into the following Corollary 6. C C 0 C C 6 C C 3 C 4 5 = C 4 C C 3 C 0 C 4 C 5 C 6 C C 4 Figre 7. A conformally indecomposable flow is decomposed nonconformally into signed-graph circit flows. Corollary 6. (a) If f is a nontrivial integral flow of an oriented singed graph, then f can be conformally decomposed into a positive integral linear combination of signed-graph circit flows. (b) Every nontrivial integral flow of an oriented signed graph can be decomposed (perhaps conformally perhaps non-conformally) into a positive integral linear combination of some signed-graph circit flows. The following proposition is trivial bt eplains why there are eactly three natral types of signed-graph circits introdced by Zaslavsky [, 4]. Proposition 7. Let Ω be a signed graph. Then the following statements are eqivalent. (a) Ω is a minimal Elerian signed graph. (b) Ω is a minimal Prime Elerian signed graph. (c) Ω is a minimal Elerian cycle-tree. (d) Ω is a signed-graph circit. Acknowledgement. The athors thank the two referees for careflly reading the manscript and offering valable sggestions and comments. References [] M. Beck and T. Zaslavsky, The nmber of nowhere-zeo flows on graphs and signed graphs, J. Combin. Theory Ser. B 96 (006), [] B. Bollabás, Modern Graph Theory, Springer, 00. 0