Transitive closure and transitive reduction in bidirected graphs
aa r X i v : . [ m a t h . C O ] J a n Transitive Closure and Transitive Reduction inBidirected Graphs
Ouahiba Bessouf a Abdelkader Khelladi a a Faculté de Mathématiques, USTHB BP 32 El Alia, Bab-Ezzouar 16111, Alger,Algérie
Thomas Zaslavsky b b Department of Mathematical Sciences, Binghamton University, Binghamton, NY13902-6000, U.S.A.
Abstract
In a bidirected graph an edge has a direction at each end, so bidirected graphsgeneralize directed graphs. We generalize the definitions of transitive closure andtransitive reduction from directed graphs to bidirected graphs by introducing newnotions of bipath and bicircuit that generalize directed paths and cycles. We showhow transitive reduction is related to transitive closure and to the matroids of thesigned graph corresponding to the bidirected graph.
Key words: bidirected graph, signed graph, matroid, transitive closure, transitivereduction
Mathematics Subject Classification 2010:
Primary 05C22, Secondary 05C20, 05C38.
Email addresses: [email protected] (Ouahiba Bessouf), [email protected] (Abdelkader Khelladi), [email protected] (Thomas Zaslavsky).
Preprint submitted to 17 January 2019
Introduction.
Bidirected graphs are a generalization of undirected and directed graphs.Harary defined in 1954 the notion of signed graph. For any bidirected graph,we can associate a signed graph of which the bidirected graph is an orienta-tion. Reciprocally, any signed graph can be associated to a bidirected graph inmultiple ways, just as a graph can be associated to a directed graph. Transitivereduction in directed graphs was introduced by A. V. Aho.The aim of this paper is to extend the concepts of transitive closure, which isdenoted by Tr ( G τ ) , and transitive reduction, which is denoted by R ( G τ ) , tobidirected graphs. We seek to find definitions of transitive closure and transi-tive reduction for bidirected graphs through which the classical concepts wouldbe a special case. We establish for bidirected graphs some properties of theseconcepts and a duality relationship between transitive closure and transitivereduction. We allow graphs to have loops and multiple edges. Given an undirected graph G = ( V, E ) , the set of half-edges of G is the set Φ( G ) defined as follows: Φ( G ) = { ( e, x ) ∈ E × V : e is incident with x } . Thus, each edge e with ends x and y is represented by its two half-edges ( e, x ) and ( e, y ) . For a loop the notation does not distinguish between its two half-edges. There is no very good notation for the two half-edges of a loop, but webelieve the reader will be able to interpret our formulas for loops.A chain (or walk ) is a sequence of vertices and edges, x , e , x , . . . , e k , x k , suchthat k ≥ and x i − and x i are the ends of e i ∀ i = 1 , . . . , k . It is elementary (or a path ) if it does not repeat any vertices or edges. It is closed if x = x k and k > . A partial graph of a graph is also known as a spanning subgraph ,i.e., it is a subgraph that contains all vertices. The terminology is due to Berge[2]. Definition 2.1 A biorientation of G is a signature of its half-edges: τ : Φ( G ) → {− , +1 } .
2t is agreed that τ ( e, x ) = 0 if ( e, x ) is not a half-edge of G ; that makes itpossible to extend τ to all of E × V , which we will do henceforth.A bidirected graph is a graph provided with a biorientation; it is written G τ =( V, E ; τ ) . Definition 2.2
An edge e = { x, y } in a bidirected graph is notated e = { x α , y β } if τ ( e, x ) = α and τ ( e, y ) = β . Two edges e, f both notated { x α , y β } are called parallel . ✉ ✉ + − x y ✉ ✉ − − x y ✉ ✉ + + x y ✉ ✉ − + x y Fig. 1. The four possible biorientations of an edge { x, y } of G τ . Each edge (including a loop) has four possible biorientations (figure 1); there-fore, the number of biorientations of G is | E | . Definition 2.3
We define two subsets of V : V +1 = { x ∈ G τ : τ ( e, x ) = +1 , ∀ ( e, x ) ∈ Φ x } is the set of source vertices ,V − = { x ∈ G τ : τ ( e, x ) = − , ∀ ( e, x ) ∈ Φ x } is the set of sink vertices , where Φ x is the set of all half-edges incident with x . (Note that this is theopposite convention for arrows to that in [13].)We observe that V +1 ∩ V − is the set of vertices that are not an end of anyedge ( isolated vertices). Definition 2.4 [3] Let G τ = ( V, E ; τ ) be a bidirected graph. Then W (resp., W ) is a function defined on V (resp., E ) as follows: W : V → Z , x W ( x ) = P e ∈ E τ ( e, x ) ,W : E → {− , , } , e W ( e ) = P x ∈ V τ ( e, x ) . Thus, W ( x ) is the number of positive half-edges incident with x less the num-ber of negative half-edges incident with x . Definition 2.5 [7] A signed graph is a triple ( V, E ; σ ) where G = ( V, E ) isan undirected graph and σ is a signature of the edge set E : σ : E → {− , +1 } .A signed graph is written G σ = ( V, E ; σ ) .3 efinition 2.6 [3] Let G σ = ( V, E ; σ ) be a signed graph and P a chain (notnecessarily elementary) connecting x and y in G σ : P : x, e , x , e , x , . . . , y, where x, x , . . . , y are vertices and e , e , . . . are edges of G . We put σ ( P ) = Y e i ∈ P σ ( e i ) . We write P α instead of P , if α = σ ( P ) . P α is called a signed chain of sign α connecting x and y . A signed chain is minimal if it contains no signed chainwith the same ends and the same sign. See figure 2. ✉ ✉ ✉ ✉ ✉✉ − − + − + x x x x yx ............................................................................................... ................ .................. ................... . ............... ............. ............. .......................................................................................................... Fig. 4. P + : x, x , x , x , x , x , y is a positive signed chain. P − : x, x , x , x , y is a negative signed chain. Fig. 2. P + contains P − as a subchain, but both P + and P − are minimal signedchains from x to y because their signs differ. Especially, a cycle in a signed graph is positive if the number of its negativeedges is even. In the opposite case, it is negative . Definition 2.7
A signed graph is balanced if all its cycles are positive [7].A signed graph is antibalanced if, by negating the signs of all edges, it becomesbalanced [8].It follows from the definitions that a cycle is balanced if, and only if, it ispositive.
Lemma 2.8
A signed graph is antibalanced if, and only if, every positive cyclehas even length and every negative cycle has odd length.
Proof.
Let G σ be a signed graph. It is antibalanced if, and only if, G − σ hasonly positive cycles. The sign of a cycle is the same in G σ and G − σ if the cyclehas even length and is the opposite if the cycle has odd length. Thus, G − σ isbalanced if, and only if, every even cycle in G σ is positive and every odd cyclein G σ is negative. Definition 2.9 [13] For a biorientation τ of a graph G τ = ( V, E ; τ ) , we definea signature σ of E , for an edge e with ends x and y , by: σ ( e ) = − τ ( e, x ) τ ( e, y ) . efinition 2.10 A signed or bidirected graph is all positive (resp., all neg-ative ) if all its edges are positive (resp., negative); i.e. in a bidirected graph,for every edge e , W ( e ) = 0 (resp., = 0 ).We observe that a bidirected graph that is all positive is a usual directedgraph.Each bidirected graph determines a unique signature. However, the numberof biorientations of a signed graph is | E | because each edge has two possiblebiorientations. Definition 2.11 [9,13] Let G τ = ( V, E ; τ ) be a bidirected graph and let X be a set of vertices of G . A new biorientation τ X of G is defined as follows: τ x ( e, x ) = − τ ( e, x ) , ∀ x ∈ X,τ x ( e, y ) = τ ( e, y ) , ∀ y ∈ V − X, for any edge e ∈ E , where x and y are the ends of the edge e . We say that thebiorientation τ X and the bidirected graph G τ X are obtained respectively from τ and G τ by switching X . If X = { x } where x ∈ V , we write τ x for simplicity.The definition of switching a signed graph is similar. Let G σ be a signed graphand X ⊆ V . The sign function σ switched by X is σ X defined as follows: σ X ( e ) = σ ( e ) , if x, y ∈ X or x, y ∈ V − X, − σ ( e ) , otherwise.We note that switching X is a self-inverse operation. It also follows from thedefinitions that the following result holds: Proposition 2.12
Let G τ be a bidirected graph and σ the signature deter-mined by τ . Let X ⊆ V . Then τ X determines the signature σ X . ✉ ✉✉ ✉ ++ + − −− − − d ca b ✲ by switching{b, c} ✉ ✉✉ ✉ ++ + − ++ − − d ca b Fig. 3. Example of switching a bidirected graph.
Proposition 2.13 (i) [12]
The result of switching a balanced signed graphis balanced. (ii) [7,12]
A signed graph is balanced if, and only if, there is a subset X ofvertices such that switching X produces a signed graph in which all edgesare positive. A signed graph is antibalanced if, and only if, there is a subset X of vertices such that switching X produces a signed graph in which alledges are negative. Proof. (i) Switching does not change the sign of any cycle.(ii) Harary [7] has shown that the set of negative edges of a balanced signedgraph, if it is not empty, constitutes a cocycle of G σ . The cocycle divides V into two sets, X and V − X , such that F consists of all edges with one endin each set. Thus by switching X , we obtain σ ( e ) = +1 ∀ e ∈ F in the newgraph G σ X and the other edge signs remain positive. Thus G σ X is all positive.Conversely, if there exists X ⊆ V such that G σ X is all positive, then G σ X isbalanced, so G σ is balanced by part (i).(iii) G σ is antibalanced ⇔ G − σ is balanced ⇔ ∃ X ⊆ V such that G ( − σ ) X = G − σ X is all positive ⇔ ∃ X ⊆ V such that G σ X is all negative.Proposition 2.13 applies to bidirected graphs (cf. [3]) because of Definition 2.9and Proposition 2.12. Similarly, all propositions about signed graphs G σ applyto bidirected graphs G τ through the signature σ determined by τ . Definition 2.14 [3] Let G τ be a bidirected graph, and let P be a chain con-necting x and y in G τ : P : xe x . . . e i x i e i +1 . . . x k − e k y. We define W P ( x i ) = τ ( e i , x i ) + τ ( e i +1 , x i ) for every x i ∈ V ( P ) , i = 1 , . . . , k − . (We note that W P ( x i ) and W P ( x j ) may differ when i = j , even if x i = x j .)Let τ ( e , x ) = α and τ ( e k , y ) = β ; then we write P = P ( α,β ) ( x, y ) : x α e x . . . e i x i +1 e i +1 . . . x k − e k y β . We call P ( α,β ) ( x, y ) an ( α, β ) bipath from x to y , or more simply a bipath from x α to y β , if:(i) k ≥ .(ii) τ ( e , x ) = α , and τ ( e k , y ) = β .(iii) W P ( x i ) = 0 , ∀ i = 1 , . . . , k − (if k > ).(iv) P ( α,β ) ( x, y ) is minimal for the properties (i)–(iii), given x α and y β .If P ( α,β ) ( x, y ) satisfies (i)–(iii), we call it a bichain from x α to y β . Thus a bipathis a minimal bichain (in the sense of (iv)); however, it need not be a path (anelementary chain). 6n the notation for a bipath P , we define x = x and x k = y . Then edge e i has vertices x i − and x i , ∀ i = 1 , , . . . , k . Definition 2.15 If P ( α,β ) ( x, y ) is a bipath from x α to y β , then P ( β,α ) ( y, x ) : y β e k x k − . . . e i +1 x i e i . . . x e x α is also a bipath, from y β to x α . It is called the reverse of P ( α,β ) ( x, y ) . Remark 2.16
In a bipath no two consecutive edges e i , e i +1 can be equal,because then by cutting out e i x i e i +1 we obtain a shorter bipath, which isabsurd. Proposition 2.17
The sign of a bichain P ( α,β ) ( x, y ) is σ ( P ) = − αβ . Proof.
Let P ( α,β ) ( x, y ) : x α e x . . . x k − e k y β be a bichain from x α to y β . Thesign of this bichain is given by σ ( P ( α,β ) ( x, y )) = Y e ∈ P ( α,β ) ( x,y ) σ ( e )= [ − τ ( e , x ) τ ( e , x )][ − τ ( e , x ) τ ( e , x )] . . . [ − τ ( e k − , x k − ) τ ( e k − , x k − )][ − τ ( e k , x k − ) τ ( e k , y )]= − τ ( e , x )[ − τ ( e , x ) τ ( e , x )] . . . [ − τ ( e k − , x k − ) τ ( e k , x k − )] τ ( e k , y ) . According to the definition of bichains we have W P ( x i ) = τ ( e i , x i ) + τ ( e i +1 , x i ) = 0 , therefore τ ( e i , x i ) τ ( e i +1 , x i ) = − ∀ i = 1 , . . . , k − . Thus, σ ( P ( α,β ) ) = − τ ( e , x ) τ ( e k , y ) = − αβ, which proves the result. Proposition 2.18
Let G τ be a bidirected graph, and let P : x e x . . . e i x i e i +1 . . . x k − e k x k , where k ≥ and e i = { x α i − i − , x β i i } for i = 1 , . . . , k , be a chain in G τ . Then P is a bipath if, and only if, (a) α i = − β i for i = 1 , . . . , k − and (b) x α i i = x α j j when i < j and ( i, j ) = (0 , k ) ;and then it is an ( α , β k ) bipath from x α to x β k k . roof. Let x = x , y = x k , α = α , β = α k . Since W P ( x i ) = β i + α i for i = 1 , . . . , k − , condition (iii) is equivalent to condition (a).Assume P is an ( α, β ) bipath from x to y . Therefore, P is a chain x α e x . . . e i x i e i +1 . . . x k − e k y β k from x α to x β k k that satisfies (i)–(iii) in Definition 2.14. If x α i i = x α j j for some i < j , then by cutting out e i +1 . . . e j we get a shorter chain with the sameproperties (i)–(iii), unless ( i, j ) = (0 , k ) . We conclude that if x i = x j ( i < j and ( i, j ) = (0 , k ) ), then α i = α j . It follows that P satisfies (b).Assume P satisfies (a) and (b). Then it satisfies (i)–(iii). Suppose P were notminimal with those properties. Then there is an ( α, β ) bipath Q from x to y whose edges are some of the edges of P in the same order as in P . If Q beginswith edge e i +1 , then it begins at x α i i and x α i i = x α = x α , therefore i = 0 by(b). Similarly, Q ends at edge e k and vertex x β k k . If Q includes edges e i and e j +1 with i < j but not edges e i +1 , . . . , e j , then x α i i = x − β i i = x α j j , contrary to(b). Therefore, Q cannot omit any edges of P . It follows that P is minimalsatisfying (i)–(iii), so P is an ( α, β ) bipath from x to y . Corollary 2.19 If P is a bipath that contains a positive cycle C , then P = C . Examples of bipaths can be seen in figure 4.We now give the different types of bipath which have a unique cycle that isnegative.
Definition 2.20 A purely cyclic bipath at a vertex x in a bidirected graph isa bipath C from x to x whose chain is a cycle. We say C is on the vertex x .The sign of C is the sign of its chain.We note that in a purely cyclic negative bipath C on x , x is the unique vertexin V ( C ) such that W C ( x ) = ± . Definition 2.21 A cyclic bipath P connecting two vertices x and y (not nec-essarily distinct) in a bidirected graph G τ , is a bipath from x to y whichcontains a unique purely cyclic bipath, which is negative. Figure 4 shows thethree possible cases. We note that α, β, γ, λ ∈ {− , +1 } . If x = y in type (a),the cyclic bipath is purely cyclic. Lemma 2.22
A cyclic bipath must have one of the forms in figure 4.
Proof.
Let P be a cyclic bipath from x to y , C the purely cyclic bipath in P , and v the vertex at which W C ( v ) = ± . The graph of P must consist of C and trees attached to C at a vertex, and it can have at most two vertices withdegree 1 because P has only two ends. We may assume P = C and y = v .8 ✉✉ ✉ ✉ ✉ ❅❅❅ (cid:3)(cid:3)(cid:3) ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ Cx x α α x - α y β (a) x α , x , . . . , x , x α , x − α , . . . , y β ✉ ✉ ✉ ✉ ✉ ✉ ✉♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ❅❅❅ (cid:3)(cid:3)(cid:3) ✉ ✉✉ ✉ ♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ Cx α v - γ - γγ γ y β (b) x α , . . . , v, . . . , v, . . . , y β ✉ ✉ ✉ ✉✉ ✉✉ ✉ ✉ ✉ ✉✉♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣✉ ❅❅❅ ❅❅❅(cid:3)(cid:3)(cid:3) (cid:3)(cid:3)(cid:3)❅❅❅ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣ ♣♣♣ Cx α w λ - λλ y β v γ - γ γ (c) x α , . . . , w, . . . , v, . . . , v, . . . , w, . . . , y β Fig. 4. The three types of cyclic bipath. In (a), x = y is possible. In (c), x or y orboth may equal w , but w = v . Since W C ( v ) = 0 , P must enter C at v (unless x = v ) and leave it at v to getto y . Therefore, there must be a tree attached to v . There cannot be a treeattached to any vertex z of C other than v , because P would have to enter thetree from C at z γ and retrace its path back to z γ in C , which would oblige P to contradict Remark 2.16. Therefore x and y are both in the tree T attachedto v , possibly with x = v . If x = v , then x must be a vertex of degree 1 in T ,or P would contradict Remark 2.16. Similarly, y must be a vertex of degree1 in T . As T must be the union of the paths in T from x and y to v , P canonly be one of the types in figure 4. Definition 2.23 [3] Let G τ be a bidirected graph, let α, β ∈ {− , +1 } , andlet C be a bipath C : x α e x . . . x k − e k x β . If α = − β , we say that C is a bicircuit of G τ .Not all bicircuits trace out a matroid circuit (which will be defined in section5). The bicircuits have been classified in a paper by Chen, Wang, and Zaslavsky[5]. 9 ✟✟✟✟✟✟✟✟✟✟✟❍❍❍❍❍❍❍❍❍❍❍❍ ✉ ✉ ✉✉ ✉ − − ++ − + − − + − + + x x x x x C : x, x , x , x , x , x , x ✉ ✉✉ ✉ ++++ x x x x C : x , x , x , x , x Fig. 5. Two kinds of bicircuit.
Let G τ be a bidirected graph. G τ is transitive if, for any ver-tices x and y (not necessarily distinct) such that there is an ( α, β ) bipath from x α to y β in G τ , there is an edge { x α , y β } in G τ . Definition 3.2
Let G τ be a bidirected graph. The transitive closure of G τ is the graph, notated Tr ( G τ ) = ( V, Tr ( E ); τ ) , such that { x α , y β } ∈ Tr ( E ) ifthere is a bipath P ( α,β ) ( x, y ) from x α to y β in G τ ( x and y are not necessarilydistinct). Remark 3.3
We see that E ⊆ Tr ( E ) . If { x α , y β } ∈ E , then { x α , y β } is theedge of a bipath of length , so { x α , y β } ∈ Tr ( E ) . Remark 3.4
If there is a bipath P ( α,β ) ( x, x ) from x α to x β in G τ , then thereis a loop { x α , x β } with sign − αβ in Tr ( G τ ) . Remark 3.5 If G τ contains a bicircuit C , then Tr ( G τ ) contains all the edges { x − α , y − β } such that { x α , y β } ∈ E ( C ) . In other words, the transitive closurecontains the opposite orientation of every edge that lies in a bicircuit in G τ .For an example see figure 9. Proposition 3.6 Tr is an abstract closure operator; that is: (i) G τ is a partial graph of Tr ( G τ ) . (ii) If H τ is a partial graph of G τ , then Tr ( H τ ) is a partial graph of Tr ( G τ ) . (iii) Tr ( Tr ( G τ )) = Tr ( G τ ) . Proof. (i) and (ii) are obvious from the definition.10iii) We prove that Tr ( G τ ) is transitive. Let P : xe x . . . e i x i e i +1 . . . x k − e k y be a bipath in Tr ( G τ ) , with e i = { x α i − i − , x β i i } for i = 1 , . . . , k . By Proposition2.18 α i = − β i for i = 1 , . . . , k − and P is an ( α , β k ) bipath. For each edge e i there is an ( α i − , β i ) bipath Q i ( x i − , x i ) in G τ (which may be e i itself). Let R = x Q x Q . . . Q k x k , the concatenation of Q , . . . , Q k . At each interme-diate vertex z of any Q i we have W R ( z ) = W Q i ( z ) = 0 by property (iii) ofDefinition 2.14. At each x i , i = 1 , . . . , k − we have W R ( x i ) = β i + α i = 0 by Proposition 2.18. Therefore, R is an ( α , β k ) bipath from x to y in G τ .We deduce that the edge { x α , y β k } is in the transitive closure of G τ . Thus, Tr ( G τ ) is transitive and its transitive closure is itself.Define W ( P ( α,β ) ( x, y )) = X e ∈ P ( α,β ) ( x,y ) W ( e ) . Theorem 3.7
Given a bidirected graph G τ and its transitive closure Tr ( G τ ) =( V, Tr ( E ); τ ) . If e = { x α , y β } is the edge in Tr ( G τ ) implied by transitive clo-sure of the bipath P ( α,β ) ( x, y ) from x α to y β in G τ , then W ( P ( α,β ) ( x, y )) = W ( e ) . Proof.
Let P ( α,β ) ( x, y ) : x α e x . . . x k − e k y β be a bipath from x α to y β in G τ .According to Definition 2.4 we have W ( P ( α,β ) ( x, y )) = τ ( e , x ) + W ( x ) + W ( x ) + . . . + W ( x k − ) + τ ( e k , y ) , and by Definition 2.14 we obtain W ( P ( α,β ) ( x, y )) = τ ( e , x ) + τ ( e k , y ) = α + β. According to Definition 2.4, W ( e ) = α + β = W ( P ( α,β ) ) .We recall that σ ( P ) designates the sign of a chain P (Definition 2.6). Corollary 3.8
Given a bidirected graph G τ and its transitive closure Tr ( G τ ) =( V, Tr ( E ); τ ) . If e = { x α , y β } is the edge implied by transitive closure of thebipath P ( α,β ) ( x, y ) from x α to y β , then σ ( P ( α,β ) ( x, y )) = σ ( e ) . Proof.
The sign of the bipath is σ ( P ( α,β ) ( x, y )) = − αβ by Proposition 2.17.We have σ ( e ) = − τ ( e, x ) τ ( e, y ) = − αβ, from which the result follows. Lemma 3.9
Let G τ be a bidirected graph and X ⊆ V . Then Tr ( G τ X ) is theresult of switching Tr ( G τ ) by X . ✉ ✉ ✉ x a b y + + P ( − , − ) ( x, y ) : x − , a, b, y − ✉ ✉ ✉ ✉ x a b y − − + + − − . ............................................... ............................................... .............................................. ............................................. ............................................ ........................................... ........................................... ............................................ ............................................. .............................................. ............................................... ................................................ ....................................................... ...................................................... ..................................................... .................................................... ................................................... .................................................. ................................................. ................................................ ................................................ ................................................. .................................................. ................................................... .................................................... ..................................................... ...................................................... ........................................................ ............................................... ............................................... .............................................. ............................................. ............................................ ........................................... ........................................... ............................................ ............................................. .............................................. ............................................... ............................................... − + + −− − Tr ( P ( − , − ) ( x, y )) Fig. 6. An example of transitive closure of a bidirected graph.
Proof.
We observe that a bipath in G τ remains a bipath after switching G τ .For a set X ⊆ V , define ι ( v ) = +1 if v / ∈ X and − if v ∈ X .Assume that e = { x α , y β } is an edge of Tr ( G τ ) , not in E ( G τ ) , that is impliedby a bipath P ( α,β ) ( x, y ) in G τ . Switch G τ and Tr ( G τ ) by X . Then e becomes { x αι ( x ) , y βι ( y ) } and P ( α,β ) ( x, y ) becomes P ( αι ( x ) ,βι ( y )) ( x, y ) . Therefore e is impliedby the bipath P ( αι ( x ) ,βι ( y )) ( x, y ) in G τ X , so e is an edge in Tr ( G τ X ) . This provesthat Tr ( G τ ) switched by X is a partial graph of Tr ( G τ X ) .By similar reasoning, if e = { x α , y β } is an edge of Tr ( G τ X ) not in E ( G τ X ) , it isimplied by a bipath P ( α,β ) ( x, y ) in G τ X . Switching by X , the edge { x αι ( x ) , y βι ( y ) } is implied by P ( αι ( x ) ,βι ( y )) ( x, y ) , which is a bipath in G τ X , so that { x αι ( x ) , y βι ( y ) } is an edge of G τ X switched by X , which is G τ . Therefore Tr ( G τ X ) switched by X is a partial graph of Tr ( G τ ) . The result follows. Proposition 3.10
The transitive closure of an all-positive bidirected graph isall positive. The transitive closure of a balanced bidirected graph is balanced.
Proof.
Assume G τ is all positive. Let P ( α,β ) ( x, y ) : x α e x . . . x k − e k y β be abipath from x α to y β ; we close this bipath by the positive edge e = { x α , y β } .Since W ( e i ) = 0 for a positive edge e i , W ( P ( α,β ) ( x, y )) = 0 . By Theorem 3.7, W ( e ) = 0 , which means that β = − α . Thus, e is positive.Assume G τ is balanced. By Proposition 2.12 there is a vertex set X ⊆ V suchthat G τ X is all positive. By the first part, Tr ( G τ X ) is all positive, therefore bal-anced, and by Lemma 3.9 it equals Tr ( G τ ) switched by X . Therefore Tr ( G τ ) equals Tr ( G τ X ) switched by X , which is balanced by Proposition 2.13.The diagram below, obtained from the results above, shows that the classicalnotion of transitive closure for directed graphs is a particular case of that12ound for bidirected graphs. G Digraph ✲ transitive closurein digraphs Tr ( G ) Digraph G τ balanced ✲ transitive closure Tr ( G τ ) Balanced ❄ G τ Positive ❄ ❄ Tr ( G τ ) Positive ❄ b y s w i t c h i n g b y s w i t c h i n g b y o r i e n t a t i o n b y o r i e n t a t i o n Definition 4.1
Let G τ = ( V, E ; τ ) be a bidirected graph. Define tr ( G τ ; H τ ) = tr ( H τ ) = the transitive closure of H τ in G τ , where H τ is a partial graph of G τ . (We can write only H τ when the larger graph, here G τ , is obvious.) Proposition 4.2
Let H τ be a partial graph of G τ . Then tr ( G τ ; H τ ) = Tr ( H τ ) ∩ G τ . Proof.
The definition implies that Tr ( H τ ) ∩ G τ ⊆ tr ( G τ ; H τ ) .Let e be an edge of tr ( G τ ; H τ ) not in H τ . The edge e is induced by a bipath P in H τ . Thus, e ∈ Tr ( E ( H τ )) . It follows that e is an edge of Tr ( H τ ) . Sincealso e ∈ E ( G τ ) , we conclude that tr ( G τ ; H τ ) ⊆ Tr ( H τ ) ∩ G τ . Definition 4.3
Let G τ = ( V, E ; τ ) be a bidirected graph. A transitive reduc-tion of G τ is a minimal generating set under tr . Thus we define R ( G τ ) =( V, R ( E ); τ ) to be a minimal partial graph of G τ with the property that tr ( G τ ; R ( G τ )) = G τ . We note that R ( G τ ) may not be unique; see Remark4.12. 13he definitions and Proposition 3.6 immediately imply that tr ( Tr ( G τ ); R ( G τ )) = tr ( Tr ( G τ ); G τ ) = Tr ( G τ ) . Proposition 4.4
Let G τ be a bidirected graph and R ( G τ ) a transitive reduc-tion. Then Tr ( R ( G τ )) = Tr ( G τ ) . Proof. Tr ( R ( G τ )) = tr ( Tr ( G τ ); R ( G τ )) = Tr ( G τ ) . Proposition 4.5
Let G τ be a bidirected graph. A partial graph H τ of G τ isa transitive reduction R ( G τ ) if, and only if, it is minimal such that G τ ⊆ Tr ( H τ ) . Proof.
It follows from Proposition 4.2 that for H τ to be a transitive reductionof G τ it is necessary that G τ ⊆ Tr ( H τ ) . It follows that H τ is a transitivereduction of G τ ⇔ H τ is minimal such that G τ ⊆ Tr ( H τ ) . Proposition 4.6 If G τ is a connected bidirected graph, then R ( G τ ) is alsoconnected. Proof.
The operator Tr does not change the connected components of a graph. Corollary 4.7
Let G τ be a bidirected graph without positive loops. If it hasno bipath of length greater than 1, then R ( G τ ) = G τ and every vertex is asource or a sink. Proof.
Since R ( G τ ) is a partial graph of G τ , it is enough to prove that eachedge e in G τ is in R ( G τ ) . Assume that there exists an edge e = { x α , y β } ∈ G τ − E ( R ( G τ )) . According to the definitions of transitive reduction and transitiveclosure, there exists a bipath from x α to y β in G τ − { e } with length k ≥ ,which is absurd.If a vertex x is neither a source nor a sink, it has incident half-edges ( e, x ) and ( f, x ) with τ ( e, x ) = +1 and τ ( f, x ) = − . Then ef is a bipath of length2, which is absurd, or e = f , which implies that e is a positive loop, which isalso absurd.We can characterize the graphs in Corollary 4.7 as follows (see figure 7): • G τ = ( V +1 ∪ V − , E ; τ ) . ( V +1 and V − are the sets of sources and sinks; seeDefinition 2.3.) • G τ is antibalanced (Definition 2.7). (Thus, if G τ is balanced, then it isbipartite.) The edges connecting a vertex of V +1 to a vertex of V − arepositive. The edges connecting two vertices of the same set are negative.14 ✉✉✉ ✉✉✉✉✉ ✉✉ ✉✉ ✉✉ ✉ PPPPP✏✏✏✏✏❆❆❆❆❆✦✦✦✦✦✦✦✦✦✟✟✟✟✟✟✟❩❩❩❩❩❩❛❛❛❛❛❛❛❛❛✑✑✑✑✑✑✑✑✑PPPPPPPPPPPPPPPPPP✏✏✏✏✏✏✏✏✏❩❩❩❩❩❩❩❩❩❅❅❅❅❅✜✜✜✜✜✜✜✜★★★★★★★★ ✬✫ ✩✪✬✫ ✩✪ + + ++ + + + + + + + + +++ + + −− − − − − − − − − − − − V +1 V − Fig. 7. The form of a graph that satisfies the hypotheses of Corollary 4.7.
Theorem 4.8 (i)
The transitive reduction R ( G τ ) is balanced if, and onlyif, G τ is balanced. (ii) R ( G τ ) is all positive if, and only if, G τ is all positive. Proof.
We conclude from Proposition 3.10 that R ( G τ ) is all positive (resp.,balanced) ⇔ Tr ( R ( G τ )) is all positive (resp., balanced) and that G τ is all pos-itive (resp., balanced) ⇔ Tr ( G τ ) is all positive (resp., balanced). By Proposi-tion 4.4, Tr ( R ( G τ )) = Tr ( G τ ) . The result follows.Theorem 4.8(ii) is important because all-positive bidirected graphs are theusual directed graphs. Thus, it says that the transitive reduction of a directedgraph, in our definition of transitive reduction, is a directed graph. The di-agram below, obtained from the results above, shows the stronger statementthat the classical notion of transitive reduction for directed graphs is a par-ticular case of our notion for bidirected graphs. G digraph ✲ transitive reductionin digraphs R ( G ) digraph G τ balanced ✲ transitive reduction R ( G τ ) balanced ❄ G τ positive ❄ ❄ R ( G τ ) positive ❄ b y s w i t c h i n g b y s w i t c h i n g b y o r i e n t a t i o n b y o r i e n t a t i o n Definition 4.9
Let
RE( G τ ) be the set of edges e such that e is in the transi-tive closure of G τ − { e } . We can say that these edges are redundant edges in15 τ . Lemma 4.10 If H τ is a partial graph of G τ , then RE( H τ ) ⊆ RE( G τ ) . Proof. If e ∈ RE( H τ ) , then e ∈ RE( G τ ) by the definition of RE .Note that two parallel edges (Definition 2.2) are both redundant. Thus, it isnecessary to exclude parallel edges in the following proposition. Proposition 4.11
For a bidirected graph without bicircuits and without par-allel edges, the graph R ( G τ ) is unique. It is obtained from G τ by removingevery redundant edge. Proof.
Since there are no parallel edges, e ∈ RE( G τ ) ⇔ e is in the transitiveclosure of a bipath of length at least .We prove first that, if e, f ∈ RE( G τ ) , then f ∈ RE( G τ − { e } ) . Suppose e = { x α , y β } is implied by a bipath P = P ( α,β ) ( x, y ) and f = { z γ , w δ } isimplied by a bipath Q = Q ( γ,δ ) ( z, w ) .If e / ∈ Q , then Q is a bipath in G τ − { e } that implies f .If e ∈ Q but f / ∈ P , then Q = Q eQ where, by choice of notation, e appearsas ( x α , y β ) in that order, so that Q = P ( γ, − α ) ( z, x ) and Q = P ( − β,δ ) ( y, w ) .Replace Q by Q P Q . This is a bichain from z γ to w δ so it contains a bipath P from z γ to w δ , in G τ − { e } , and P implies f .If e ∈ Q and f ∈ P , then Q = Q eQ where e , Q and Q are as inthe previous case, and P = P f P where f appears in P as either ( z γ , w δ ) or ( w δ , z γ ) . Suppose the first possibility. Then P = P ( α, − γ ) ( x, z ) so P Q is a bichain from x α to x − α ; therefore P Q contains a bicircuit, which isimpossible. Now suppose the second possibility and let P ∗ denote the reverseof the bipath P (Definition 2.15). Then P Q P ∗ Q ∗ is a bichain from x α to x − α ; therefore it contains a bicircuit, which is impossible. Therefore, this casecannot occur.We conclude that f ∈ RE( G τ − { e } ) for every edge f ∈ RE( G τ ) , f = e .Therefore, RE( G τ − { e } ) ⊇ RE( G τ ) − { e } . Since G τ − { e } is a partial graphof G τ , RE( G τ ) − { e } ) ⊆ RE( G τ ) so RE( G τ − { e } ) = RE( G τ ) − { e } . Byinduction, RE( G τ − RE( G τ )) = RE( G τ ) − RE( G τ ) = ∅ . We also concludethat f ∈ Tr ( G τ − { e } ) and by induction that RE( G τ ) ⊆ Tr ( G τ − RE( G τ )) .Therefore, R ( G τ ) = G τ − RE( G τ ) . This is unique.Figure 8 shows that the transitive closure of the bipath P ( − , − ) (2 ,
3) : 2 − , , − contains the edge { − , − } which is a redundant edge.16 ✉✉ ✁✁✁✁✁✁ ❆❆❆❆❆❆ − − − − + − G τ ✉ ✉✉ ✁✁✁✁✁✁ ❆❆❆❆❆❆ − − + − R ( G τ ) Fig. 8. { − , − } is a redundant edge. Remark 4.12
We show an example in which the transitive reduction is unique,and an example in which it is not unique. If C and C are two symmet-rical bicircuits, that is, { x α , y β } ∈ E ( C ) ⇔ { x − α , y − β } ∈ E ( C ) ), then Tr ( C ) = Tr ( C ) = Tr ( G τ ) . Hence in figure 9 G τ has only one transitivereduction, C , but Tr ( G τ ) has both C and C as transitive reductions. ✉ ✉✉ ✁✁✁✁✁✁ ❆❆❆❆❆❆ − − + − ......................................................................................................... ................. ................. ................... ..................... ........................ ........................... + + G τ ✉ ✉✉ ✁✁✁✁✁✁ ❆❆❆❆❆❆ − − + − ......................................................................................................... ................. ................. ................... ..................... ........................ ........................... . ............................ ......................... ...................... ................... ................. ................. ....................................................................................................................... ......................... ....................... ...................... .................... .................. ................. .................. .................. ................... .................... + + − − − + Tr ( G τ ) ✉ ✉✉ ✁✁✁✁✁✁ ❆❆❆❆❆❆ − − + − C = R ( G τ ) ✉ ✉✉ ✁✁✁✁✁✁ ❆❆❆❆❆❆ − − + + − + C = R ( Tr ( G τ )) Fig. 9. Example for Remark 4.12. C is an R ( G τ ) and an R ( Tr ( G τ )) . C is an R ( Tr ( G τ )) , but not an R ( G τ ) because it is not contained in G τ . For legibility, in Tr ( G τ ) we do not show the positive loops that exist at every vertex. Let G τ be a bidirected graph and E = { e , e , . . . , e n } its set of edges. Assume E is linearly ordered by a linear ordering < in index order, i.e., e i < e j ⇔ i < j .Let ( G τ ) i be a family of graphs constructed from G τ as follows: ( G τ ) = G τ and ( G τ ) i = ( G τ ) i − − e i if e i = { x α , y β } is implied by transitive closureof a bipath P ( α,β ) ( x, y ) in ( G τ ) i − − { e i } , ( G τ ) i − otherwise.We put S < ( G τ ) = { e i ∈ E : e i ∈ ( G τ ) i − and e i / ∈ ( G τ ) i } . Proposition 4.13
Let G τ be a bidirected graph. For each linear ordering < of E , G τ − S < ( G τ ) is a transitive reduction of G τ . Conversely, if R ( G τ ) =( V, R ( E ); τ ) is a transitive reduction of G τ , then R ( G τ ) = G τ − S < ( G τ ) forsome linear ordering < of E . roof. Assume a linear ordering < of E . By construction, if e i ∈ S < ( G τ ) ,then e i ∈ Tr (( G τ ) i ) . Let m = | E | ; then Tr ( G τ ) = Tr m ( G τ − S < ( G τ )) (the m -times iterate of Tr ) = Tr ( G τ − S < ( G τ )) by Proposition 3.6. By Proposition4.2, tr ( G τ ; G τ − S < ( G τ )) = G τ since G τ ⊆ Tr ( G τ ) = Tr ( G τ − S < ( G τ )) . If G τ − S < ( G τ ) were not a minimal partial graph that generates G τ under Tr ,then there would be an edge e j ∈ E − S < ( G τ ) such that e j is implied by abipath P ( α,β ) ( x, y ) in G τ − S < ( G τ ) − e j . This bipath is in ( G τ ) j − − e j so byconstruction e j / ∈ ( G τ ) j , therefore e j ∈ S < ( G τ ) , which is absurd. Therefore G τ − S < ( G τ ) is a transitive reduction of G τ .Suppose R ( G τ ) is a transitive reduction of G τ . Let S = E − E ( R ( G τ )) . Everyedge in S is implied by a bipath in R ( G τ ) . Linearly order E by < so that S = { e , . . . , e k } is initial in the ordering. Then at step i ≤ k of the constructionof S < ( G τ ) , edge e i is implied by a bipath in ( G τ ) i − − { e i } so e i ∈ S < ( G τ ) ; butat step i > k , ( G τ ) i = R ( G τ ) , which has no such bipath because of minimalityof R ( G τ ) . Corollary 4.14
Let G τ be a bidirected graph and < a linear ordering of E ( G τ ) . If S < ( G τ ) = ∅ , then R ( G τ ) = G τ . Corollary 4.15 If P ( α,β ) ( x, y ) is a bipath, then R ( P ( α,β ) ( x, y )) = P ( α,β ) ( x, y ) . Proof.
For the graph P ( α,β ) ( x, y ) , we have S < ( G τ ) = ∅ . In this section we study the relationship between transitive closure and tran-sitive reduction.
Proposition 4.16
Let G τ be a bidirected graph. Then every transitive reduc-tion of G τ is a transitive reduction of Tr ( G τ ) . Proof.
We apply Proposition 4.13. Let R ( G τ ) be a transitive reduction of G τ .Choose a linear ordering of E ( Tr ( G τ ) in which the edges of Tr ( G τ ) − E ( G τ ) are initial and the edges of R ( G τ ) are final. By the definition of Tr , the m edges of Tr ( G τ ) − E ( G τ ) are in S < ( Tr ( G τ ) − E ( G τ )) and ( Tr ( G τ )) m = Tr ( G τ ) .The proposition follows.It may not be true that every transitive reduction of Tr ( G τ ) is an R ( G τ ) . Let H τ be a bidirected graph that has more than one transitive reduction, andlet G τ = R ( H τ ) . Then G τ = R ( G τ ) . Since H τ ⊆ Tr ( H τ ) = Tr ( G τ ) , everytransitive reduction of H τ is a transitive reduction of Tr ( G τ ) , but only oneof those transitive reductions can be G τ = R ( G τ ) . That cannot happen if G τ has no bicircuit. We prove a lemma first.18 emma 4.17 Let G τ be a bidirected graph without a bicircuit or parallel edges.Then Tr ( G τ ) has no bicircuit. Proof.
Suppose e = { x α , y β } ∈ Tr ( G τ ) − E ( G τ ) . That means there is abipath P ( α,β ) ( x, y ) in G τ . Now suppose there is a bicircuit C from z γ to z − γ in G τ ∪ { e } , C = C eC , where we may assume C ends at x − α and C beginsat y − β . Then C P C is a closed bichain from z γ to z − γ in G τ , so it containsa bicircuit, but that is absurd. Therefore, G τ ∪ { e } contains no bicircuit. Theproof follows by induction on the number of edges in Tr ( G τ ) − E ( G τ ) . Proposition 4.18
Let G τ be a bidirected graph without a bicircuit or paralleledges. Then R ( Tr ( G τ )) = R ( G τ ) . That is, the unique transitive reduction of Tr ( G τ ) is the (unique) transitive reduction of G τ . Proof.
By Lemma 4.17, Tr ( G τ ) has no bicircuit. According to Proposition4.11, Tr ( G τ ) has a unique transitive reduction. R ( G τ ) is such a transitivereduction. Therefore, R ( Tr ( G τ )) = R ( G τ ) . Corollary 4.19
Let G τ be a bidirected graph without a bicircuit or paralleledges. If S < ( G τ ) = ∅ , then R ( Tr ( G τ )) = G τ . Proof. If S < ( G τ ) = ∅ , then according to Corollary 4.14 we have R ( G τ ) = G τ .Moreover, it follows from Proposition 4.18 that R ( Tr ( G τ )) = R ( G τ ) , fromwhich the result follows. Proposition 4.20
Let G τ be a bidirected graph without a bicircuit or paralleledges. Then Tr ( R ( Tr ( G τ ))) = Tr ( G τ ) . Proof.
By Proposition 4.18, R ( Tr ( G τ )) = R ( G τ ) ⇒ Tr ( R ( Tr ( G τ ))) = Tr ( R ( G τ )) ⇒ Tr ( R ( Tr ( G τ ))) = Tr ( G τ ) by Proposition 4.4. We indicate by b ( G τ ) the number of balanced connected components of G τ . Theorem 5.1 [12]
Given a signed graph G σ , there is a matroid M ( G σ ) asso-ciated to G σ , such that a subset F of the edge set E is a circuit of M ( G σ ) if,and only if, either Type (i) F is a positive cycle, or Type (ii) F is the union of two negative cycles, having exactly one commonvertex, or Type (iii) F is the union of two vertex-disjoint negative cycles and an ele-mentary chain which is internally disjoint from both cycles. he rank function is r ( M ( G τ )) = | V | − b ( G τ ) . This matroid is now called the frame matroid of G σ . See figure 10, where werepresent a positive (resp., negative) cycle by a quadrilateral (resp., triangle). ✉✉ ✉✉ ◗◗◗◗◗ ◗◗◗◗◗✑✑✑✑✑✑✑✑✑✑ Type (i) ✑✑✑✑✑✑✑✑✑◗◗◗◗◗◗◗◗◗ ✉ ✉ ✉✉ ✉
Type (ii) ✑✑✑✑✑◗◗◗◗◗ ◗◗◗◗◗✑✑✑✑✑ ✉✉ ✉ ✉ ✉✉
Type (iii)
Fig. 10.
The matroid associated to the bidirected graph is the matroid associated toits signed graph (given by Definition 2.9).
Definition 5.2 ([9]; [10, Def. 1.1 of §3.1]) A signed graph G σ is called quasi-balanced ( m-balanced in [9,10]) if it does not admit circuits of types (ii) and(iii). We have the same definition for bidirected graphs. Proposition 5.3
A connected signed graph G σ is quasibalanced if, and onlyif, for any two negative cycles C and ´ C we have | V ( C ) ∩ V ( ´ C ) | ≥ . Proof.
Sufficiency results from Definition 5.2 and Theorem 5.1.To prove necessity, suppose that G σ admits two negative cycles C and ´ C .Suppose | V ( C ) ∩ V ( ´ C ) | = 0 . Since G σ is connected, there exists a chainconnecting a vertex of C with a vertex of ´ C , therefore there exists a circuit oftype (iii) which contains C and ´ C . Suppose | V ( C ) ∩ V ( ´ C ) | = 1 . Then C ∪ ´ C isa circuit of type (ii). Both cases are impossible; therefore | V ( C ) ∩ V ( ´ C ) | > . Problem 5.4
Describe all quasibalanced signed graphs, i.e., signed graphsin which every pair of negative cycles has at least two common vertices. (Cf.[14].)
Proposition 5.5 If G τ is a quasibalanced bidirected graph, then R ( G τ ) isquasibalanced. Proof. As M ( G τ ) is without circuits of type (ii) and (iii), and since R ( G τ ) isa partial graph of G τ , the result follows. Proposition 5.6
Let G τ be a bidirected graph such that R ( G τ ) is quasibal-anced. Let < be a linear ordering of E . If G τ is quasibalanced, then for everyedge e belonging to the set S < ( G τ ) , e is not in the transitive closure of anycyclic bipath in G τ . roof. Let G σ be the signed graph corresponding to G τ .Assume that there is an edge e = { x α , y β } , belonging to the set S < ( G τ ) , whichis in the transitive closure of the cyclic bipath P = P ( α,β ) ( x, y ) , containing thenegative cycle C . This implies that P ∪ { e } is a circuit of the type (ii) or (iii)of G σ , if the cycle ´ C of P ∪ { e } that contains e is negative. By Proposition2.17 the sign of P is − αβ , which is also the sign of e . Therefore, the sign ofthe closed chain P e is + . This sign equals σ ( C ) σ ( ´ C ) , so σ ( ´ C ) = σ ( C ) = − .Thus G τ is not quasibalanced, which is absurd.We do not have a sufficient condition for quasibalance. The converse of Propo-sition 5.6 is false. Consider G τ with V = { , , , , , , } and edges e = 1 − + , + + , + + , + + , − + , − + , − + , + + , − + , linearly ordered in that order. We claim that e is redundant using the path P : 15672 , and that no other edge is redundant. There is no matroid circuitof type (ii) or (iii) in G τ − e . But G τ − − + is a matroid circuit of type (ii).Therefore, G τ is not quasibalanced, but R ( G τ ) = G τ − e is quasibalanced.However, e is not in the transitive closure of any cyclic bipath. P and e arethe only bipaths from to .Let F denote the closure of F in a matroid. Lemma 5.7
Let G τ be a bidirected graph with edge set E . If e ∈ E − R ( E ) ,then e belongs to the closure R ( E ) in M ( G τ ) . If e ∈ E ( Tr ( G τ )) − E , then e belongs to the closure E in M ( Tr ( G τ )) . Proof.
Let P be a bipath in R ( G τ ) which induces e ∈ E . Then P ∪ { e } is amatroid circuit of type (i), (ii) or (iii). Thus, e ∈ R ( E ) in M ( G τ ) .The second statement follows from the first because in M ( Tr ( G τ )) , E is theclosure of R ( E ) , which equals R ( E ) . Theorem 5.8
Let G τ be a bidirected graph. Then r ( M ( Tr ( G τ ))) = r ( M ( G τ )) = r ( M ( R ( G τ ))) . Proof.
For r ( M ( Tr ( G τ ))) = r ( M ( G τ )) , it is enough to use Lemma 5.7.For r ( M ( R ( G τ ))) = r ( M ( Tr ( G τ ))) , it is enough to cite Proposition 4.4 andreplace G τ in the previous case by R ( G τ ) .The definitions of a connected matroid in [11] apply to the matroids of signedgraphs. In particular: 21 efinition 5.9 Let G σ = ( V, E ; σ ) be a signed graph. The matroid M ( G σ ) is connected if each pair of distinct edges e and é from G σ , is contained in acircuit C of M( G σ ). Theorem 5.10
Let G τ = ( V, E ; τ ) be a bidirected graph and let R ( G τ ) beany transitive reduction of G τ . If M ( R ( G τ )) is connected, then M ( G τ ) isconnected. Proof.
Theorem 5.8 implies that E ( R ( G τ )) = E ( G τ ) = E ( Tr ( G τ )) in M ( Tr ( G τ )) .It follows by standard matroid theory, since M ( R ( G τ )) is connected, that M ( G τ ) and M ( Tr ( G τ )) are connected.We note that the converse is false. For example, let P ( α,β ) ( x, y ) be a bipathof length not less than 2, whose graph is an elementary chain, and let e bethe edge { x α , y β } . Let G τ = P ( α,β ) ( x, y ) ∪ { e } . Then P ( α,β ) ( x, y ) = R ( G τ ) ,but M ( P ( α,β ) ( x, y )) is disconnected while M ( G τ ) is connected (since the cor-responding signed graph is a positive cycle). References [1] Gautam Appa and Balazs Kotnyek, A bidirected generalization of networkmatrices, Networks 47(4) (2006), 185–198.[2] Claude Berge,
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