Lobachevskii Journal of Mathematics

http://ljm.ksu.ru
ISSN 1818-9962

Vol. 24, 2006, 3–12 \begin{matrix}c \\ ◯ \end{matrix} H. Azanchiler

H. Azanchiler

EXTENSION OF LINE-SPLITTING OPERATION FROM GRAPHS TO BINARY MATROIDS

(submitted by M. M. Arslanov)

Abstract. In this paper, we characterize the n-line splitting operation of graphs in terms of cycles of respective graphs and then extend this operation to binary matroids. In matroids, we call this operation an element-set splitting. The resulting matroid is called the es-splitting matroid. We characterize circuits of an es-splitting matroid. We also characterize the es-splitting matroid in terms of matrices. Also, we show that if M is a connected binary matroid then the es-splitting matroid M_{X}^{e} is also connected.

1. Introduction

In [9], Slater specified the n-point splitting and n-line splitting operations in graphs in the following way:

Let G be a graph and U be a vertex of G with deg U ≥ 2n − 2. Let G_{1} be the graph obtained from G by replacing U by two adjacent vertices U_{1} and U_{2}, and if vertex X is adjacent to U in G, written X adj U, then make X adj U_{1} or X adj U_{2} (but not both) such that deg U_{1} ≥ n and deg U_{2} ≥ n (see Figure 1.1).

Key words and phrases. Binary matroid, n-line splitting, element-set splitting, connected matroid.

[figure: X_{1}; X_{2}; X_{k}; U; Y_{1}; Y_{2}; G; Y_{1}; Y_{2}; X_{1}; X_{2}; X_{k}; a; U_{1}; U_{2}; Y_{h}; Y_{h}; Figure 1.1; G_{1}]

We say that G_{1} is obtained from G by n-point splitting operation. The operation of n-line splitting is defined as follows.

Let G be a graph and e = UV be an edge of G with deg U ≥ 2n − 3 with U adjacent to X_{1}, X_{2}, …, X_{k}, Y_{1}, Y_{2}, …, Y_{h}, where h and k ≥ n−2. Let H be the graph obtained from G by replacing U by two adjacent vertices U_{1} and U_{2} with V adj U_{1}, V adj U_{2}, U_{1} adj X_{i} and U_{2} adj Y_{j}, where 1 ≤ i ≤ k and 1 ≤ j ≤ h and deg U_{1} ≥ n, deg U_{2} ≥ n. Let H be said to arise from G by n-line splitting (see Figure 1.2). We also say that H is a n-line splitting of G.

[figure: X_{1}; X_{2}; X_{k}; U; V; Y_{1}; Y_{2}; G; Y_{1}; Y_{2}; X_{1}; X_{2}; X_{k}; V; a; U_{1}; U_{2}; e; γ; H; Figure 1.2; Y_{h}; Y_{h}]

The above operations can be related to the earlier splitting operations. Let X_{i}U = x_{i} (1 ≤ i ≤ k) and UV = e and U_{1}U_{2} = a, U_{2}V = γ. Moreover, let X = {x_{1}, x_{2}, ⋯, x_{k}, e} and Δ = {e, a, γ}. We denote H by G_{X}^{e}. If G_{X} denote the splitting of G with respect to X (see [6]), then G_{X}^{e} = G_{X}+a+γ. If G′_{X} is obtained from G by n-point splitting operation, then G_{X}^{e} = G′_{X}+γ. In other words, G_{X}^{e}\{a, γ} = G_{X} and G_{X}^{e}\{γ} = G′_{X}.

In the next proposition, we characterize the cycles of the graph G_{X}^{e} in terms of the cycles of G.

In [2] Raghunathan, Shikare and Waphare generalized the splitting operation of graphs to binary matroids. Shikare, Azadi and Waphare [6, 7] extended the notions of n-point splitting from graphs to binary matroids. The authors in [4] determined the bases of splitting matroids and Shikare, Azanchiler and Waphare [8] characterized cocircuits of splitting matroids. We extend the n-line splitting from graphs to binary matroids in next Section. Also we characterize circuits of the result matroid. For the matroid theory we refer the reader to [1, 10]. Proposition 1.1. Let G be a graph and e = uv be an edge of G with deg u ≥ 2n − 3 and u adjacent to X_{1}, X_{2}, ⋯, X_{k}, Y_{1}, Y_{2}, ⋯, Y_{h}, where h and k ≥ n − 2. Let X_{i}u = x_{i} and Y_{j}u = y_{j} for 1 ≤ i ≤ k; 1 ≤ j ≤ h. Let X = {e, x_{1}, x_{2}, ⋯, x_{k}} and Y = {y_{1}, y_{2}, ⋯, y_{h}}. Then C is a cycle in G_{X}^{e} if and only if C satisfies one of the following conditions:

(1) C is a cycle in G, containing precisely two elements of X.

(2) C is a cycle in G containing no element of X.

(3) C = C_{1} ∪ C_{2}, where C_{1} and C_{2} are edge-disjoint cycles of G, each contains exactly one edge of X and C_{1} ∪ C_{2} contains, no cycle of type (1) or (2).

(4) C = C_{1} ∪ {a} when C_{1} is a cycle of G, containing precisely one edge of X.

(5) C = C_{1} ∪ {e, γ}, where C_{1} is a cycle of G, containing exactly one element of X − {e}.

(6) C = (C_{1}\e)∪{a, γ}, where C_{1} is a cycle of G, containing precisely one element of X − e and the edge e.

(7) C = (C_{1} \ e) ∪ {γ}, where C_{1} is a cycle of G, containing the edge e of X.

(8) C = {e, a, γ}.

2. Splitting of a Binary Matroid with Respect to an Element and a Set

Now, we extend the notion of n-line splitting operation from graphs to binary matroids. In the first step, we consider matrix approach to this operation in binary matroids. Definition 2.1. Let M be a binary matroid on a set S and X be a subset of S, e ∈ X. Suppose that A is a matrix over GF (2), that represents the matroid M. Let A_{X}^{e} be the matrix that is obtained by adjoining an extra row to A with this row being zero everywhere except in the columns corresponding to the elements of X where it takes the value 1, and then adjoining two columns a and γ to the resulting matrix such that the column a is zero everywhere except in the last row (new row) where it takes the value 1, and γ is a sum of two column vectors corresponding to a and e.

Let M_{X}^{e} be the vector matroid of the matrix A_{X}^{e}. We say that M_{X}^{e} has been obtained from M by splitting e and X in M. The transition from M to M_{X}^{e} is called splitting of M with respect to e and X. For convenience, we say that M_{X}^{e} is a element -set splitting (es-splitting) matroid.

Remark 2.2. Let r and r′ be the rank functions of M and M_{X}^{e}, respectively. Then r′(M_{X}^{e}) = r(M) + 1.

In the next proposition we characterize the circuits of the matroid M_{X}^{e}.

Theorem 2.3. Let M = (S, C) be a binary matroid, X ⊆ S, e ∈ X and a, γ ∉ S. Then M_{X}^{e} = (S ∪{a, γ}, C_{X}^{e}), where C_{X}^{e} = C_{0} ∪C_{1} ∪C_{2} ∪C_{3} ∪{Δ} with Δ = {e, a, γ} and C_{0} = {C ∈ C | C contains an even number of elements of X}; C_{1} = The set of minimal members of {C_{1} ∪ C_{2} | C_{1}, C_{2} ∈ C, C_{1} ∩ C_{2} = ϕ and each C_{1} and C_{2} contains an odd number of element of X such that C_{1} ∪ C_{2} contains no member of C_{0}}; C_{2} = {C ∪{a} | C ∈ C and C contains an odd number of element of X}. C_{3} = { C ∪ {e, γ}, | C ∈ C, e ∉ C and C contains odd number of elements of X}. ∪ { (C \ e) ∪ {γ} | C ∈ C, e ∈ C and C contains odd number of elements of X }. ∪ { (C \ e) ∪ {a, γ} | C ∈ C, e ∈ C and C \ e contains odd number of elements of X }. Proof. We prove that C_{X}^{e} satisfies the circuit axioms of a matroid.

(1) Let A, B ∈ C_{X}^{e}, we show that if A ⊆ B, then A = B. In other words, we prove that A ⊄ B and B ⊄ A. The property clearly holds, if both A and B belong to C_{0}, C_{1} or C_{2}.

Now, assume that both A and B belong to C_{3} and let A = C_{1} ∪ {e, γ} and B = (C_{2} \ e) ∪ {γ}, where C_{1}, C_{2} ∈ C, C_{1} and C_{2} contain an odd number of element X, e ∉ C_{1} and e ∈ C_{2}. Thus e ∈ A and e ∉ B, so it follows that A ⊄ B. Similarly, a ∉ A and a ∈ B, implies that B ⊈ A.

Next, let A ∈ C_{i} and B ∈ C_{j}, i ≠ j and i, j = 0, 1, 2, 3.

(i) Let A ∈ C_{0} and B ∈ C_{1}. Then B = C_{1} ∪ C_{2}, where C_{1} and C_{2} are circuits of M, satisfying the properties stated in the Definition 4.2.1 of C_{1}.

(ii) Let A ∈ C_{0} and B ∈ C_{2}. Then B = B_{1} ∪ {a}, where B is a circuit of M, containing an odd number of element of X. Since A ⊄ B_{1} and B_{1} ⊄ A, it follows that A ⊄ B and B ⊄ A, as desired.

(iii) Suppose A ∈ C_{0} or C_{1} and B ∈ C_{3}. Since γ ∈ B and γ ∉ A, then

A ⊄ B and B ⊄ A. (iv) Let A ∈ C_{1} and B ∈ C_{2}. Suppose A = C_{1} ∪ C_{2} and B = B_{1} ∪ {a},

where C_{1}, C_{2} and B_{1} are circuits of M, containing an odd number of element of X. Since a ∈ B and a ∉ A, B ⊄ A. Also A ⊄ B, for A ⊆ B implies that A ⊆ B_{1} and C_{1} ∪ C_{2} ⊂ B_{1}, leads to a contradiction.

(v) Let A ∈ C_{2} and B ∈ C_{3}. Then A = A_{1} ∪ {a} and B = B_{1} ∪ {e, γ} or B = (B_{2} \e)∪{γ}, where A_{1}, B_{1} and B_{2} are circuits in M each contains an odd number of elements of X and e ∉ B_{1}, e ∈ B_{2}. Since γ ∈ B but γ ∉ A, B ⊄ A. Similarly, A ⊄ B.

(2) Let A, B ∈ C_{X}^{e} and A ≠ B. We prove that there exists D ∈ C_{X}^{e} such that D ⊆ AΔB.

Firstly, let A, B ∉ C_{2} ∪ C_{3} ∪ {Δ}. By binarity of M,

AΔB = C′_{1} ∪ C_{2} ∪ ⋯ ∪ C′_{m},  (∗)

where C′_{1}, C′_{2}, ⋯ and C′_{m} are disjoint circuits of M. Since A and B each contains an even number of element of X, AΔB contains an even number of element of X. If AΔB contains no element of X, then each circuit C′_{i}, i = 1, 2, ⋯, m is a member of C_{0} and is contained in AΔB. If for some j, 1 ≤ j ≤ m, C′_{j} contains an even number of element of X, then D = C′_{j}. Otherwise, m must be an even integer and for every j = 1, 2, ⋯, m, C′_{j} must contain an odd number of element of X. If C′_{1} ∪ C′_{2} contains a member of C_{0}, say C, then we take D = C. Otherwise; D = C′_{1} ∪ C′_{2} or a minimal member of C_{1} contained in it.

Secondly, suppose A, B ∈ C_{2} ∪ C_{3} ∪ {Δ}, then we have the following cases:

(I) Let A, B ∈ C_{2} and A = A_{1} ∪ {a}, B = B_{1} ∪ {a}. Then AΔB = A_{1}ΔB_{1} = C′_{1}∪C′_{2}∪⋯∪C′_{m}, where C′_{1}, C′_{2}, ⋯ and C′_{m} are disjoint circuits of M. Since each of A and B contains an odd number of elements of X, AΔB contains an even number of elements of X. By similar argument as above, we can find D ∈ C_{X}^{e} such that D ⊆ AΔB.

(II) Let A and B belong to C_{3}, we have the following subcases:

(i) A = A_{1} ∪ {e, γ} and B = B_{1} ∪ {e, γ}. (ii) A = (A_{1} \ e) ∪ {γ} and B = (B_{1} \ e) ∪ {γ}. (iii) A = A_{1} ∪ {e, γ}, B = (B_{1} \ e) ∪ {γ}. (iv) A = A_{1} ∪ {e, γ}, B = (B_{1} \ e) ∪ {a, γ}. (v) A = (A_{1} \ e) ∪ {a, γ} and B = (B_{1} \ e) ∪ {a, γ}. (vi) A = (A_{1} \ e) ∪ {a, γ}, B = (B_{1} \ e) ∪ {a, γ}.

In cases (i), (ii) and (vi), we have AΔB = A_{1}ΔB_{1} = C′_{1} ∪C′_{2} ∪⋯∪C′_{m}, where C′_{1}, C′_{2}, ⋯, C′_{m} are disjoint circuits of M. By the similar arguments as in (1), we can find D ∈ C_{X}^{e} such that D ⊆ AΔB. In case (iii), we have AΔB = (A_{1}ΔB_{1}) ∪ {a}.

Let A_{1}ΔB_{1} = C′_{1} ∪ C′′_{2} ∪ ⋯ ∪ C′′_{n}, where C′′_{1}, C′′_{2}, ⋯, C′′_{n} are disjoint circuits of M. If C′′_{1} contains an even number of elements of X, then it will be an element of C_{X}^{e} which is contained in A_{1}ΔB_{1}, and hence in AΔB. If C′′_{1} contains an odd number of elements of X, then C′′_{1} ∪ {a} is an element of C_{2}, contained in AΔB. By similar argument, the cases (iv) and (v) follow.

(III) Let A ∈ C_{2} and B ∈ C_{3}. Then A = A_{1} ∪ {a} and for B, we have two subcases: (1) B = B_{1} ∪ {e, γ}, and (2) B = (B_{1} \ e) ∪ {γ}. Thus, in (1), we have AΔB = (A_{1}ΔB_{1})∪{e, a, γ}, where A_{1} and B_{1} are circuits of M, containing an odd number of element of X. Clearly, A_{1}ΔB_{1} ⊆ AΔB. Let A_{1}ΔB_{1} = C′_{1} ∪ C′_{2} ∪ C′_{m}, where C′_{1}, C′_{2}, ⋯, C′_{m} are disjoint circuits of M. By similar arguments as in (2), we can find D ∈ C_{X}^{e} such that D ⊆ A_{1}ΔB_{1} and hence D ⊆ AΔB. In (2), we have A = A_{1} ∪ {a} and B = (B_{1} \ e) ∪ {γ}. So AΔB = [(A_{1}ΔB_{1}) \ e] ∪ {a, γ}, where A_{1} and B_{1} are circuits in M. By the argument as given above, we can find D ∈ C_{X}^{e}, such that D ⊆ AΔB.

(IV) Let A = Δ and B ∈ C_{2}. Then B = B_{1} ∪{a}, where B_{1} is a circuit in M containing an odd number of element of X. We have two subcases:

(i) e ∉ B_{1}. Then AΔB is a circuit of M, containing an even number of elements of X. Thus D = AΔB. (ii) e ∈ B_{1}. Then AΔB is also a circuit in M_{X}^{e} and so D = AΔB.

(V) Let A = Δ and B ∈ C_{3}. Then B = B_{1}∪{e, γ} or B = (B_{1}\e)∪{γ}, where B_{1} is a circuit of M, containing an odd number of element of X. Since AΔB_{1} ⊂ AΔB, we can find D = Δ, where D ∈ C_{X}^{e} such that D ⊆ AΔB_{1}, and hence D ⊆ AΔB.

We conclude that C_{X}^{e} is a collection of circuits of a binary matroid on the set S ∪ {a, γ}.

□

Definition 2.4. With notation as above, we call the matroid (S ∪ {a, γ}, C_{X}^{e}) as the es-splitting of M = (S, C) and denote it by M_{X}^{e}. Thus M_{X}^{e} = (S′, C_{X}^{e}), where S′ = S ∪ {a, γ}.

Example 2.5. Consider the matroid M = F_{7} with ground set S = {1, 2, 3, 4, 5, 6, 7} and the set of circuits

C = {{1, 2, 4}, {1, 3, 6}, {2, 6, 7}, {4, 5, 6}, {1, 5, 7}, {3, 4, 7},
{2, 3, 5}, {3, 5, 6, 7}, {2, 4, 5, 7}, {1, 3, 4, 5}, {1, 2, 3, 7},
{2, 3, 4, 6}, {1, 2, 5, 7}, {1, 4, 6, 7}}

Let X = {1, 2, 4} and e = 1. Then the circuit set of M_{X}^{e} is

C_{X}^{e} = {{3, 5, 6, 7}, {2, 4, 5, 7}, {1, 3, 4, 5}, {1, 2, 3, 7}, {2, 3, 4, 6},
{1, 2, 5, 6}, {1, 4, 6, 7}, {1, 3, 6, a}, {1, 2, 4, a}, {2, 6, 7, a},

{4, 5, 6, a}, {1, 5, 7, a}, {3, 4, 7, a}, {2, 3, 5, a}, {1, 2, 6, 7, γ}, {1, 4, 5, 6, γ}, {1, 3, 4, 7, γ}, {1, 2, 3, 5, γ}, {2, 4, 8}, {3, 6, γ}, {5, 7, γ}, {1, a, γ}, {3, 4, 5, a, γ}, {2, 3, 7, a, γ}, {2, 5, 6, a, γ},

{4, 6, 7, a, γ}}.

Theorem 2.6. The matrix A_{X}^{e} represents the splitting matroid M_{X}^{e}.

3. Connectedness of the Splitting Matroid M_{X}^{e}

In [5] Shikare characterized the connectedness in splitting matroidM_{X}. The next theorem characterize connectedness of M_{X}^{e}. Theorem 3.1. Let M = (S, C) be a binary connected matroid. Then M_{X}^{e} is connected. Proof. Let M be a connected matroid on S. Then for every pair x, y ∈ S there is a circuit of M containing x and y. We show that for any two elements x and y belonging to S ∪ {a, γ}, there is a circuit of M_{X}^{e} containing x and y. We consider the following cases:

(1) Let x, y ∈ {a, γ}. Then x, y ∈ Δ and we are through.

(2) Suppose x, y ∉ {a, γ}. Then x, y ∈ S and M has a circuit, say C

containing x and y. We have the following two subcases:

(i) C contains an even number of elements of X. Then we are through.

(ii) C contains an odd number of elements of X. Then C ∪ {a} is a circuit of M_{X}^{e} containing x and y.

(3) Let x = a and y ∈ S. Then there is a circuit C ∈ C such that

y, e ∈ C. We have two subcases:

(i) Suppose C contains an even number of elements of X. Then C ∈ C_{X}^{e}, e ∈ C ∩ Δ, y ∈ C and a ∈ Δ. By [3], there is a circuit say C′ of M_{X}^{e} such that a, y ∈ C′.

(ii) Let C contains an odd number of elements of X. Then C ∪ {a} is a circuit in M_{X}^{e} and a, y ∈ C ∪ {a}.

(4) Let x = γ and y ∈ S. Then there is a circuit C ∈ C such that

y, e ∈ C. We consider the following two subcases:

(i) C contains an even number of element of X. Then C ∈ C_{X}^{e}, e ∈ C ∩ Δ, y ∈ C and γ ∈ Δ, so by [3], there is a circuit of M_{X}^{e} containing γ and y.

(ii) C contains an odd number of element of X. Then (C \ e) ∪ {γ} is a circuit of M_{X}^{e} containing γ and y. □

Remark 3.2. Converse of Theorem 3.1 is not true. For example, let M be a cycle matroid of the graph G (See Figure 3.1). Let X = {e, x_{1}, x_{2}}. Then M_{X}^{e} is a cycle matroid of G_{X}^{e}.

[figure: x_{1}; x_{2}; w; y; e; z; u; x_{1}; x_{2}; z; w; e; a; y; u_{1}; u_{2}; γ; G_{X}^{e}; G; Figure 3.1]

We observe that the matroid M = M(G) is disconnected while M_{X}^{e} = M(G_{X}^{e}) is connected.

The next theorem shows that one can obtain a connected matroid from disconnected matroid with the help of es-splitting operation.

Theorem 3.3. Let M be a bridgeless binary matroid on S with n components M_{1}, M_{2}, ⋯, M_{n}. Let x_{i} be chosen from M_{i} for i = 1, 2, ⋯, n and X = {x_{1}, x_{2}, ⋯, x_{n}}, e ∈ X. Then M_{X}^{e} is a connected matroid on S ∪ {a, γ}.

Proof. Let M_{1} = (S_{1}, D_{1}), M_{2} = (S_{2}, D), ⋯, M_{n} = (S_{n}, D_{n}) be the n components of M, where D_{i} denote the collection of circuits of M_{i}. Then S_{i} ∩ S_{j} = ϕ for i ≠ j, i, j = 1, 2, ⋯, n and ∪_{i=1}^{n}S_{i} = S. X = {x_{1}, x_{2}, ⋯, x_{n}}, where x_{i} ∈ S_{i} for i = 1, 2, ⋯, n. Since e ∈ X, there is a j such that x_{j} = e, where 1 ≤ j ≤ n. Suppose A_{i} = {C ∈ D_{i} | x_{i} ∉ C} for i = 1, 2, ⋯, n and A_{n+1} = {C_{i} ∪ C_{j} | C_{i} ∈ D_{i}, C_{j} ∈ D_{j}, x_{i} ∈ C_{i}, x_{j} ∈ C_{j}, i ≠ j and i, j = 1, 2, ⋯, n}. Let B_{i} = {C ∪ {a} | C ∈ D_{i}, x_{i} ∈ C, i = 1, 2, ⋯, n} and

C_{i} = {E ∪ {e, γ} | E ∈ D_{i}, e ∉ E, x_{i} ∈ E}
∪{(F \ e) ∪ {γ} | F ∈ D_{i}, e ∈ F, x_{i} ∉ F \ {e}}
∪{(G \ e) ∪ {a, γ} | G ∈ D_{i}, e ∈ G, x_{i} ∈ G \ {e}}.

The matroid M_{X}^{e} has the circuit set C_{X}^{e}, where C_{X}^{e} = A_{1} ∪ A_{2} ∪ ⋯ ∪ A_{n} ∪ A_{n+1} ∪ B_{1} ∪ B_{2} ∪ ⋯ ∪ B_{n} ∪ C_{1} ∪ C_{2} ∪ ⋯ ∪ C_{n} ∪ {Δ}. Claim. For every pair of elements α, β of S ∪ {a, γ} there is a circuit of M_{X}^{e}, containing α and β. We have the following cases:

(I) Let α, β ∈ S. Then we consider the following subcases:

(i) Suppose α and β belong to one component, say M_{i}. Then there is a circuit say C_{i} of M_{i}, containing α and β. If x_{i} does not belong to C_{i}, then C_{i} is a circuit of M_{X}^{e}, containing α and β. If x_{i} ∈ C_{i}, then for any j ≠ i, C_{j} is a circuit of M_{j} containing x_{j}. Thus C_{i} ∪ C_{j} is a circuit of M_{X}^{e} containing α and β.

(ii) Let α and β belongs to different components of M, say α ∈ M_{i} and β ∈ M_{j}. Consider the circuits C_{i} and C_{j}, where C_{i} contains α and x_{i} and C_{j} contains β and x_{j}. Then C_{i} ∪ C_{j} is a required circuit of M_{X}^{e}.

(II) Let α ∈ S and β ∈ {a, γ}. Then α ∈ S_{i} for some i. Consequently, there is a circuit of M_{i}, say C_{i}, containing α and x_{i}. If β = a then C_{i} ∪ {a} is a circuit of M_{X}^{e} containing α and β. If b = γ, then (C_{i} \ {e}) ∪ {γ} is a circuit of M_{X}^{e} containing α and β.

(III) If α = a and β = γ, then α, β ∈ Δ, a 3-circuit in M_{X}^{e}. □

References

[1] Oxley J. G., Matroid Theory, Oxford University Press, New York, (1992).

[2] Raghunathan T. T., Shikare M. M. and Waphare B. N., Splitting in a binary matroid, Discrete Mathematics, 184 (1997), 261-271.

[3] Recski A., Matroid Theory and Its Applications, Springer Verlag, Berlin (1989).

[4] Shikare M. M., Azadi G., Determination of the bases of a splitting matroid, European Journal of Combinatorics, 24 (2003), 45-52.

[5] Shikare M. M., Splitting operation and connectedness in binary matroids, Indian J. Pure and Appl. Math., 31 (12) (2000), 1691-1697..

[6] Shikare, M. M., Azadi G. and Waphare B. N., Generalized splitting operation and its applications to binary matroids (Preprint).

[7] Shikare, M. M. and Azadi, G., Element splitting operation for binary matroids (preprint).

12 H. AZANCHILER

[8] Shikare, M. M., Azanchiler, H., Waphare, B. N., The cocircuits of splitting matroids. (Preprint)

[9] Slater P. J., Soldering and point splitting, J. Combinatorial Theory, 24(3) (1978), 338-343.

[10] Welsh D. J. A., Matroid Theory, Academic Press, London (1976).

Department of Mathematics, University of Urmia, Urima–57135 (Iran) E-mail address: azanchiler@yahoo.com

Received August 12, 2006