Cobiased graphs: Single-element extensions and elementary quotients of graphic matroids

Zaslavsky (1991) introduced a graphical structure called a biased graph and used it to characterize all single-element coextensions and elementary lifts of graphic matroids. We introduce a new, dual graphical structure that we call a cobiased graph and use it to characterize single-element extensions and elementary quotients of graphic matroids.


Introduction
An elementary lift of a matroid M is a matroid of the form N \e 0 in which N/e 0 = M and e 0 is neither a loop nor coloop of N .Single-element coextensions and elementary lifts of graphic matroids were characterized in terms of graphic structures by Zaslavsky [10,11].Aside from the work in [10,11], single-element coextensions and elementary lifts of graphic matroids have been objects of consistent interest in matroid theory and related fields.Guenin's investigation [2] into integral polyhedra related to binary elementary lifts of graphic matroids is notable.
An elementary quotient of a matroid M is a matroid of the form N/e 0 in which N \e 0 = M and e 0 is neither a loop nor coloop of N .Elementary quotients of graphic matroids have also been of consistent interest, in particular binary elementary quotients.Guenin's result in [2] applies not only to binary elementary lifts of graphic matroids but also binary elementary quotients.Guenin, Pivotto, and Wollan [3] explored the relationships between binary elementary lifts and quotients of graphic matroids.Seymour's original proof of the decomposition theorem for regular matroids [9] uses binary single-element extensions and elementary quotients of graphic matroids.In the field of error-correcting codes, Hakimi and Bredeson [4] constructed binary codes using circuit spaces of binary single-element and multiple-element extensions and quotients of graphic matroids.Jungnickel and Vanstone [5] do the same with q-ary codes.
Despite all of the interest in single-element extensions and elementary quotients of graphic matroids, there has been no general description of them.Recski characterized connected single-element extensions and elementary quotients of graphic matroids [7,8] that are representable over a given field and made some generalizations.In this paper we will fully characterize all single-element extensions and elementary quotients of graphic matroids in terms of graphical structures.

Cobiased Graphs
Let G be a graph and X ⊂ V (G) or X ⊂ G.The coboundary of X is denoted by δ(X) and is the set of links (i.e., non-loop edges) of G with exactly one endpoint in X.Consider a tripartition {X 1 , X 2 , X 3 } of the vertices of one connected component of G into nonempty subsets such that each induced subgraph G[X i ] is connected and there is at least one edge of G connecting each pair of these three subgraphs.The union of the three bonds (i.e., minimal edge cuts) B 1 = δ(X 1 ), B 2 = δ(X 2 ), and B 3 = δ(X 3 ) is called a tribond (see Figure 1).
A linear class of bonds of G is a subset L of the set of all bonds in G satisfying the property that every tribond contains zero, one, or three bonds from L; that is, a tribond cannot contain exactly two bonds from L. We call the pair (G, L) in which G is a graph and L a linear class of bonds a cobiased graph.Bonds in L are called cobalanced and bonds not in L are called un-cobalanced.(The name comes from the fact that cobalance of bonds is dual to balance of cycles in [10] et seq.)A linear class of bonds L is trivial when all bonds are cobalanced; that is, L is the set of all bonds of G.

Join and Complete Join Matroids of Cobiased Graphs
In this section we will describe two matroids associated with a cobiased graph (G, L).These matroids are denoted by J 0 (G, L) and J(G, L) and are called respectively the complete join and join matroids of (G, L).The term "join" is used to echo the considerable amount of literature on T -joins of graphs that are dependent sets of binary elementary quotients of graphic matroids.The matroids J 0 (G, L) and J(G, L) are defined in Section 3.1 in terms of their cocircuits using Crapo's Theorem [1, p. 62] on single-element extensions of matroids.Crapo's Theorem also immediately implies that J 0 (G, L) and J(G, L) characterize respectively single-element extensions and elementary quotients of graphic matroids.From there we will determine the bases, independent sets, rank functions, and circuits of these two matroids.We will also provide a graphical description of deletions and contractions.

Cocircuits
A pair of bonds B 1 , B 2 in a graph G is a modular pair when the number of connected components of G − (B 1 ∪ B 2 ) is two more than the number of connected components of G. Thus B 1 , B 2 form a modular pair of bonds when: B 1 ∪ B 2 forms a tribond, B 1 ∪ B 2 form the configuration in Figure 1 but with no edges between X 1 and X 2 (see Figure 2, left), or B 1 and B 2 are in two distinct connected components of G (see Figure 2, right).When B 1 , B 2 is a modular pair of bonds but B 1 ∪ B 2 is not a tribond (i.e., B 1 ∪ B 2 is one of the structures from Figure 2) we will call B 1 ∪ B 2 a dibond.Note that a modular pair of bonds share an edge if and only if their union is a tribond.The matroid of Theorem 3.1 is J 0 (G, L), the complete join matroid of (G, L).If L is trivial, then the single-element extension associated with L is just M (G) along with a new element that is either a loop or coloop.Theorem 3.1.If L is a non-trivial linear class of bonds of G, then there is a matroid with element set E(G) ∪ e 0 in which e 0 is neither a loop nor a coloop and whose cocircuits consist of the following: (1) bonds in L, (2) sets of the form B ∪ e 0 in which B is a bond not in L, and (3) tribonds and dibonds which do not contain a bond from L.
Conversely, if N is a single-element extension of the graphic matroid M (G) with new element e 0 which is neither a loop nor a coloop of N , then there is a non-trivial linear class of bonds L of G such that the cocircuits of N consist of the sets above.
Proof.This follows the cocircuit version of Crapo's Theorem [1, p.62] on single-element extensions of matroids as long as modularity of bonds as we have defined them graphically is exactly how modular pairs of cocircuits behave in the graphic matroid M (G).This will complete our proof.
Let B 1 and B 2 be distinct bonds in G and let and only if the flat H 1 ∩ H 2 has 2 more connected components than does G, which is how we defined a modular pair of bonds.
The join matroid of a cobiased graph (G, L) is defined as J(G, L) = J 0 (G, L)/e 0 .If L is trivial, then J(G, L) = M (G).Theorem 3.2 is an immediate corollary of Theorem 3.1.Theorem 3.2.If L is a non-trivial linear class of bonds in G, then the cocircuits of J(G, L) consist of the following: (1) bonds in L and (2) tribonds and dibonds which do not contain a bond from L.
Furthermore, if N is an elementary quotient of a graphic matroid M (G), then N = J(G, L) for some non-trivial linear class L.

Bases and independent sets
Consider a partition π = {X 1 , . . ., X k } of V (G) into nonempty parts such that each induced subgraph G[X i ] is connected.Denote the set of such partitions for G by Lattice(G).The partial ordering is the usual refinement partial ordering; that is, given two such partitions Think of the edges of ∪G[X i ] as a relation on V (G) and extend it to an equivalence relation ≡ π .If τ ∈ Lattice(G), extend ≡ π ∪ ≡ τ to an equivalence relation ≡; then the join π ∨ τ is the partition that corresponds to ≡. Supposing that π, τ ∈ Lattice(H), this formula for the join is the same whether viewed in H or in G.) In particular Lattice(G) is a join subsemilattice of Lattice(K n ), which is the usual partition lattice of the set V (K n ). Given ] by Interior(π), or Interior G (π) when necessary.The collection of such edge sets is of course exactly the set of flats of the graphic matroid M (G).The set of edges of G which are not in Interior(π) is denoted by Exterior(π) or Exterior G (π).Note that Exterior(π) is a union of bonds.
Given a cobiased graph (G, L), we call π ∈ Lattice(G) cobalanced with respect to L when every bond in Exterior(π) is cobalanced; otherwise the partition is un-cobalanced.The maximal element of Lattice(G) is π G , that is, the partition of V (G) given by the connected components of G itself.The coatoms of Lattice(G), that is, the elements of Lattice(G) which are covered by π G , are those partitions π for which Exterior(π) is a bond.Theorem 3.3.If L is a non-trivial linear class of bonds of G, then the bases of J 0 (G, L) consist of the following: (1) edge sets of maximal forests of G and (2) sets of the form F ∪ e 0 in which F is a maximal forest with one edge deleted such that the bond Exterior(π F ) is un-cobalanced.
When one edge is deleted from a maximal forest, one component tree is broken into two trees.The bond between those two trees is Exterior(π F ).
Proof.For a general matroid M , B is a basis if and only if B is a minimal set which intersects every cocircuit (see, for example, [6, p.77]).Now since edge sets of cycles are dependent in M (G), they are also dependent in its single-element extension J 0 (G, L).So if B is a base of J 0 (G, L), then B\e 0 is the edge set of a forest in G. Theorem 3.1 now implies the following: e 0 / ∈ B if and only if B is a maximal forest and e 0 ∈ B if and only if B\e 0 is obtained from a maximal forest by the deletion of one edge e such that the bond Exterior(π F ) is un-cobalanced.
Theorem 3.4.If L is a non-trivial linear class of bonds of G, then the bases of J(G, L) consist of the forests F for which Exterior(π F ) is an un-cobalanced bond.
Proof.Because L is non-trivial, e 0 is not a loop or coloop of J 0 (G, L).Thus the bases of J(G, L) are obtained from the bases of J 0 (G, L) that contain e 0 by removing e 0 .The result now follows from Theorem 3.3.Theorem 3.5.If L is a non-trivial linear class of bonds of G, then the independent sets of J 0 (G, L) consist of the following: (1) Edge sets of forests.
(2) Edge sets of the form F ∪ e 0 in which F is a forest and π F is un-cobalanced.
Proof.This follows from Theorem 3.3 because (1) and (2) describe exactly the subsets of the bases of J 0 (G, L).Theorem 3.6.If L is a non-trivial linear class of bonds of G, then the independent sets of J(G, L) consist of the edge sets of forests F such that π F is un-cobalanced.
Proof.This follows from Theorem 3.4 because these are exactly the subsets of the bases of J(G, L).

Rank
Proof.Part (1) follows from the fact that the rank of X in M (G) and its single-element extension J 0 (G, L) must be the same.Theorem 3.5 implies that there is a circuit containing e 0 in the set X ∪ e 0 if and only if π X is cobalanced.This implies Parts (2) and (3).
Proof.This follows from Theorem 3.7 and the fact that J(G, L) = J 0 (G, L)/e 0 .

Circuits
If J is a forest in (G, L) for which π J is cobalanced and J is minimal with respect to this property, then J is called an L-join of the cobiased graph (G, L).Being an L-join means that deleting any edge of J creates a bond that is not in L. Theorem 3.9.If L is a non-trivial linear class of bonds of G, then the circuits of J 0 (G, L) consist of the following: (1) edges sets of the form J ∪ e 0 in which J is an L-join and (2) edges sets of cycles.
Proof.Edge sets of cycles are circuits of M (G) and therefore are also circuits in J 0 (G, L).Hence, any other circuit consists of the edge set of some forest along with e 0 .Suppose that F is the edge set of a forest for which F ∪ e 0 is a circuit.Theorem 3.5 implies that F ∪ e 0 is independent when π F is un-cobalanced and dependent when π F is cobalanced.Thus F ∪ e 0 is a circuit when π F is cobalanced and F is minimal with respect to this property.Thus F is an L-join.Theorem 3.10.If L is a non-trivial linear class of bonds of G, then the circuits of J(G, L) consist of the following: (1) edge sets of L-joins and (2) edge sets of cycles that do not contain L-joins.
Proof.This follows from Theorem 3.9 and the fact that J(G, L) = J 0 (G, L)/e 0 .

Deletions and contractions
Let G be a graph and e a link in G.The bonds of G/e are the bonds of G which do not contain e. Define (G, L)/e = (G/e, L/e) in which L/e is the set of all bonds B ∈ L which do not contain e.Now (G/e, L/e) is a cobiased graph because any tribond of G/e is a tribond of G.
The situation for deletions is only slightly more complicated.If B is a bond in G\e, then either B or B ∪ e is a bond in G.We define (G, L)\e = (G\e, L\e) in which L\e is the set of all bonds B in G\e for which either B or B ∪ e ∈ L. Now (G\e, L\e) is a cobiased graph because if T is a tribond of G\e, then either T or T ∪ e is a tribond of G. ∈ D, the partition of V (G) associated with D is the same for G and G\e.Thus π D is still a cobalanced partition in Lattice(G\e), so D is a dependent set of J(G\e, L\e).Conversely, if D is a dependent set of J(G\e, L\e), then either D contains a cycle, in which case it is dependent in [J(G, L)]\e, or D is a forest of G\e such that π D ∈ Lattice(G\e) is cobalanced.Again, the partition of V (G) associated with D is the same in G as in G\e, so π D ∈ Lattice(G) is cobalanced, from which it follows that D is dependent in [J(G, L)]\e.
(1) The proof is similar to that of Part (2) with only the added detail of noting the presence of e 0 in dependent sets without cycles.(3 and 4) These follow a similar strategy to the proofs of (1) and (2) but by comparing cocircuits.The details are left to the reader.

Vertex union
An operation that preserves graphic matroids is the union of two graphs at a single vertex.This operation has the same property for cobiased graphs and join matroids, as we see in Theorem 3.12.That fact is hinted at within discussions in [8] but is not fully developed.Theorem 3.12.
Proof.The set of bonds in G 1 is exactly the set of bonds in G 2 ; furthermore, the tribonds of G 1 are the same as the tribonds of G 2 because no tribond can have edges in more than one block of a graph.This proves (1).Part (2) follows from these same facts and comparing cocircuits.The linear class L a,b has a special property: In every tribond the number of un-cobalanced bonds is even.This looks like a dual of the characteristic property of biased graphs derived from edge signs (gains in the 2-element group), that in every theta graph the number of unbalanced cycles is even (called "additive bias" in [10]).By analogy, let us call such a linear class of bonds additive.Signed graphs are especially simple (and important) gain graphs.That raises the following two questions: Is there similar importance for additive cobias?Is there a simple general construction of additively cobiased graphs, dual in some sense to the construction of additively biased graphs from signed graphs?Example 4.9 shows that Z 2 does in fact relate to the class L a,b , which is a step in the direction of answers.

Two simple examples
Example 3.15.More generally let W ⊆ V (G) have even cardinality, define a bond to be cobalanced if it separates W into two even subsets, and let L + W be the linear class of all such bonds.Then L + W is also additive.
Both examples are forms of quotient labeled cobias; see Example 4.9.

Gain graphs and linear classes of bonds
An oriented edge in a graph is an edge e along with a chosen direction along that edge.If e is an oriented edge, then the reverse orientation is denoted by −e when using additive notation and by e −1 when using multiplicative notation.(We will be using additive notation except in Sections 4.2 and 4.3.)The set of all possible oriented edges in G is denoted by ⃗ E(G).An oriented bond ⃗ B is a bond B = δ(X) along with a choice of orientation for the edges of B, either all away from X or all towards X.The reverse orientation of ⃗ B is denoted by − ⃗ B when using additive notation and ⃗ B −1 when using multiplicative notation.
Let Γ be a group.A Γ-gain graph is a pair (G, φ) in which G is a graph and φ : E(G) → Γ is a mapping such that φ(−e) = −φ(e) for an additive group and φ(e −1 ) = φ(e) −1 for a multiplicative group.(All our additive groups are abelian.Our multiplicative groups are not assumed to be abelian.)The function φ is called a Γ-gain function.Gain graphs give rise to biased graphs (see [10]).Similarly, they give rise to cobiased graphs, though not always.Proof.Consider a tribond containing the three bonds δ(X 1 ), δ(X 2 ), δ(X 3 ) and assume without loss of generality that δ(X 1 ), δ(X 2 ) ∈ L φ .Let ⃗ B 1 and ⃗ B 2 be the oriented bonds obtained from δ(X 1 ) and δ(X 2 ) by orienting all edges away from X 1 and X 2 .Let ⃗ B 3 be the oriented bond obtained from δ(X 3 ) by orienting all of its edges towards

Cobiased graphs from additive gain graphs
which implies that a tribond cannot have exactly two bonds in L φ , which is our result.

Cobiased planar graphs using gains over arbitrary groups
Let Γ be a multiplicative group (not necessarily abelian), let G be a connected graph embedded in the plane, and let (G, φ) be a Γ-gain graph.Consider an oriented bond ⃗ B in G.The oriented edges of ⃗ B correspond to a closed walk in the topological dual graph G * .Thus there is a well-defined cyclic ordering of the edges of ⃗ B up to a choice of a starting edge and clockwise or counterclockwise direction.So, given such an oriented bond ⃗ B, let e 1 , . . ., e k be a cyclic ordering with e 1 as the starting edge.Define φ( ⃗ B) = φ(e 1 ) • • • φ(e k ).Note that any other choice of starting edge yields a product φ(e i ) • • • φ(e k )φ(e 1 ) • • • φ(e i−1 ), which is conjugate to φ(e 1 ) • • • φ(e k ) in the group Γ.Furthermore, a different choice of direction yields a product that is the inverse of the original.Therefore, φ( ⃗ B) = 1 for any one choice of starting edge and direction if and only if φ( ⃗ B) = 1 for all possible choices of starting edge and direction.Say that a bond B is cobalanced when φ( ⃗ B) = 1 for some choice of starting edge and direction and let L φ be the set of cobalanced bonds given by φ.Proposition 4.2.If G is a graph embedded in the plane, Γ is a multiplicative group, and Proof.This proof is similar to the one for Proposition 4.1, but with the added concern of picking starting edges and directions for each bond in a tribond to match with the others.

An example not realizable by gains
The example here is essentially the topological dual of [

Cycle shifting
Let (G, φ) be a Γ-gain graph in which Γ is an additive group.Let C be a cycle in G with oriented edges e 1 , . . ., e k in cyclic order in one direction around C. For any a ∈ Γ, let ψ C,a be the Γ-gain function on G for which φ(e i ) = a and φ(e) = 0 for all e such that ±e / ∈ {e 1 , . . ., e k }.Now for any oriented bond ⃗ B, φ( ⃗ B) = (φ+ψ C,a )( ⃗ B).Thus L φ = L φ+ψ C,a .We call this operation shifting.We say that two Γ-gain functions φ 1 and φ 2 (or two Γ-gain graphs (G, φ 1 ) and (G, φ 2 )) are shifting equivalent when φ 1 is obtained from φ 2 via a sequence of shifts.Note that this relation is an equivalence relation.In order to prove Theorem 4.3, we need a concept of normalizing gains.So if (G, φ) is a Γ-gain graph and T is a maximal forest in G, then we say that φ is T -normalized when φ is zero on each edge of G not in T .
It might seem that quotient labelings are different than gains; however, they are actually equivalent constructions.Theorem 4.6 describes how to get gains from a quotient labeling and Theorem 4.8 describes how to get a quotient labeling from gains.
Given a Γ-quotient labeling π of G and a maximal forest T in G, define a T -normalized Γ-gain function φ π,T as follows.For each oriented edge e not in T , let φ π,T (e) = 0.For an edge e in T , let B = δ(X) be the bond Exterior(π T \e ) and say that ⃗ B is the orientation of B directed towards X.Orient e towards X as well.Now set φ π,T (e) = π(X).Proof.One shift operation can be performed in each connected component of G. Thus the result is true if and only if it is true for connected graphs, so we may assume that G is connected.Consider the following well-known operation on spanning trees, which we will call edge exchange in this proof.If T is a spanning tree of G and e / ∈ T , then for any edge f ̸ = e on the unique cycle in T ∪ e, (T \f ) ∪ e is a spanning tree of G.It is well known that if G is a connected graph and T and T ′ are spanning trees of G, then there is a sequence of spanning trees T 1 , . . ., T k ⊆ (T ∪ T ′ ) such that T = T 1 , T ′ = T k , and T i+1 is obtained from T i by an edge exchange.So to complete the proof it suffices to show that φ π,T i+1 is obtained from φ π,T i by a single shift operation.
Say that T i+1 = (T i \f ) ∪ e and let g / ∈ {e, f } be any other edge in T i .Thus T i \{f, g} has exactly three connected components S 1 , S 2 , S 3 as shown in Figure 4. Orient edges f and g as indicated.There are two cases for the placement of e relative to f and g as shown in the figure.Let C be the unique cycle in T i ∪ e oriented in the opposite direction to e and f .Note that g is in C in the right configuration of Figure 4 but not in the left configuration.Now let

Deletions and contractions for gains
Let φ be a Γ-gain function.The set L φ /e is defined in Section 3.5 as the set of bonds in L φ that do not contain e and it is shown that L φ /e is a linear class of bonds of G/e.Let φ/e be the Γ-gain function defined on G/e by restriction of φ to E(G/e) = E(G)\e.Proposition 4.10 is immediate.

Deletions and contractions for quotient labelings
Let π be a Γ-quotient labeling of G and let e be a link in G with endpoints u and v. Let w be the vertex obtained by the contraction of e in G. Define π/e to be the labeling on V (G/e) given by (π/e)(x) = π(x) when x ∈ V (G) ∩ V (G/e) and (π/e)(w) = π(u) + π(v).
Proposition 4.12.If π is a Γ-quotient labeling of G and e is a link in G, then π/e is a Γ-quotient labeling of G/e and L π /e = L π/e .

Gains and quotient labelings using fields
Let F be any field.Denote the additive group of F by F + .Scalar multiplication of F + -gain functions and quotient labelings using a nonzero element of F does not affect the resulting linear class of bonds.The proof of Proposition 4.14 is evident.Readers who are familiar with partial fields will note that this operation generalizes immediately to partial fields.(2) aπ is a quotient labeling of G with L aπ = L π .

Figure 2 :
Figure 2: Every dibond is of exactly one of two possible types.

Theorem 3 . 11 .
If (G, L) is a cobiased graph and e is a link in G, then (1) [J 0 (G, L)]\e = J 0 (G\e, L\e), (2) [J(G, L)]\e = J(G\e, L\e), (3) [J 0 (G, L)]/e = J 0 (G/e, L/e), and (4) [J(G, L)]/e = J(G/e, L/e).Proof.(2) We compare dependent sets.Let D ⊆ E(G).If D is a dependent set of [J(G, L)]\e, then e / ∈ D and either D contains a cycle or D is a forest such that π D ∈ Lattice(G) is a cobalanced partition.If D contains a cycle, then it is a dependent set in J(G\e, L\e).If D is a forest, then because e /

Example 3 .
13. Pick vertices a, b in G. Define a bond as cobalanced when it does not separate a and b.Let L a,b denote this set of cobalanced bonds.Proposition 3.14.(G, L a,b ) is a cobiased graph.Proof.Consider a tribond corresponding to a tripartition {X, Y, Z} of V (G).If a, b are in the same part, all three bonds are cobalanced.If a ∈ X and b ∈ Y , then the bond δ(Z) is cobalanced and the other two bonds are not.That is, an even number of the three are un-cobalanced.It follows that L a,b is a linear class of bonds.

10 ,
Example 5.8].Consider the labeled graph G ∼ = K 2,4 shown on the left in Figure 3 with all edges oriented in the downward direction.Let L = {a 1 a 2 a 3 a 4 , a 1 a 2 b 3 b 4 , b 1 b 2 a 3 a 4 }.Up to reembedding of G, a tribond in G is of one of the two types shown on the right of Figure 3.Note that any such tribond contains at most one bond from L. Thus (G, L) is a cobiased graph.

a 1 a 2 a 3 a 4 b 1 b 2 b 3 b 4 v 1 v 2 Figure 3 :
Figure 3: All edges are oriented in the downward direction.Every tribond is of one of the two types shown.

Theorem 4 . 6 . 6 . 4 . 7 .
Let Γ be an additive group and π a Γ-quotient labeling of graph G.If T is a maximal forest of G, then for every oriented bond⃗ B in G, φ π,T ( ⃗ B) = π( ⃗ B).Proposition 4.7 is necessary for the proof of Theorem 4.Proposition Let Γ be an additive group and π a Γ-quotient labeling of graph G.If T and T ′ are maximal forests in G, then φ π,T and φ π,T ′ are shifting equivalent.

Figure 4 :
Figure 4: Figure for the proof of Proposition 4.7.

Proposition 4 . 10 .
If φ is a Γ-gain function and e is an edge of G, then L φ /e = L φ/e .The set L φ \e is defined in Section 3.5 as the set of bonds B in G\e for which either B or B ∪ e is a bond in L φ .As long as e is not an isthmus of G, there is a Γ-gain function ψ on G that is shifting equivalent to φ and for which ψ(e) = 0. (See Proposition 4.4.)Define ψ\e to be the Γ-gain function on G\e defined by restriction of ψ to E(G\e) = E(G)\e.Proposition 4.11.If φ is a Γ-gain function and e is a non-isthmus edge of G, then there is ψ that is shifting equivalent to φ such that ψ(e) = 0 and L φ \e = L ψ \e = L ψ\e .Proof.The existence of ψ is implied by Proposition 4.4.Now, if B ∈ L ψ\e , then (ψ\e)( ⃗ B) = 0. Since B is a bond of G\e, there is a bond B e ∈ {B, B ∪ e} of G. Since ψ(e) = 0, we get φ( ⃗ B e ) = ψ( ⃗ B e ) = 0.This implies that B e ∈ L φ , which implies that B ∈ L φ \e.Conversely, if B ∈ L φ \e, there is a bond B e ∈ {B, B ∪ e} of L φ for which ψ( ⃗ B e ) = φ( ⃗ B e ) = 0.This implies that (ψ\e)( ⃗ B) = 0, which makes B ∈ L ψ\e .

Proposition 4 .
14.If (G, φ) is an F + -gain graph, π an F + -quotient labeling, and a a nonzero element of F, then (1) L φ = L aφ and . A bond B of G/e is in L π /e if and only if B is a bond of G not containing e and B ∈ L π if and only if both endpoints of e are in X or both endpoints of e are not in X where B = δ(X) if and only if B is a bond of G/e in L π/e .For Proposition 4.13, we define the vertex labeling π\e on G\e by π\e = π.Proposition 4.13.If π is a Γ-quotient labeling of G and e is a link in G, then π\e is a Γ-quotient labeling of G\e if and only if e is not an un-cobalanced isthmus of (G, L π ).Furthermore, if e is not an un-cobalanced isthmus of (G, L π ), then L π \e = L π\e .Proof.The first statement is obvious.Now if e is not an un-cobalanced isthmus, consider a bond B in G\e and let B e ∈ {B, B ∪ e} be the corresponding bond in G. Now B ∈ L π \e if and only B e ∈ L π if and only if B ∈ L π\e .