On n-connected minors of the es-splitting bi- nary matroids

The es-splitting operation on an n-connected binary matroid may not yield an n-connected matroid for (n ≥ 3). In this paper, we show that given an n-connected binary matroid M of rank r, the resulting es-splitting binary matroid has an n-connected minor of rank-(r+ 1) having |E(M)|+ 1 elements.


Introduction
Slater [13] specified the n-line splitting operation on graphs as follows. Let G be a graph and e = uv be an edge of G with deg u ≥ 2n − 3 with u adjacent to v, x 1 , x 2 , . . . , x k , y 1 , y 2 , . . . , y h , where k and h ≥ 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 (1 ≤ i ≤ k), and u 2 adj y j (1 ≤ j ≤ h), where deg u 1 ≥ n and deg u 2 ≥ n. The transition from G to H is called an n-line splitting operation. We also say that H is obtained from G by an n-line splitting operation. This construction is explicitly illustrated with the help of Figure 1.
Slater [13] proved that if G is n-connected and H is obtained from G by n-line-splitting operation, then H is n-connected. In fact, he characterized 4-connected graphs, in terms of the 4-line www.ejgta.org On n-connected minors of the es-splitting binary matroids | P. P. Malavadkar et al. splitting operation along with some other operations. The notion of connectivity of graphs also has been studied in [6,13] and connectivity of binary matroids has been studied in [3,14]. Suppose G is a graph with n vertices and m edges. Let X = {e, x 1 , x 2 , . . . , x k } be a subset of E(G). The incident matrix A of G is a matrix of size n × m. The row corresponding to the vertex u has 1 in the columns of e, x 1 , x 2 , . . . , x k , y 1 , y 2 , . . . , y h and 0 in the other columns. The graph H has (n + 1) vertices and (m + 2) edges. The incidence matrix A of H is a matrix of size (n + 1) × (m + 2). The row corresponding to u 2 has 1 in the columns of y 1 , y 2 , . . . , y h , γ and 0 in the other columns, where as the row corresponding to the vertex u 1 has 1 in the columns of e, x 1 , x 2 , . . . , x k and 0 in other columns. One can check that the matrix A can be obtained from A by adjoining an extra row corresponding to the vertex u 1 to A with entries zero every where except in the columns corresponding to e, x 1 , x 2 , . . . , x k where it takes the value 1. The row vector obtained by addition (mod 2) of row vectors corresponding to vertices u and u 1 will corresponds to the row vector of the vertex u 2 in A .
Noticing the above s Azanchiler [1] extended the notion of n-line-splitting operation from graphs to binary matroids in the following way: Definition 1. Let M be a binary matroid on a set E and let X be a subset of E with e ∈ X. Suppose A is a matrix representation of M over GF (2). Let A e X be a matrix obtained from A by adjoining an extra row δ X to A with entries zero every where except in the columns corresponding to the elements of X, where it takes the value 1 and then adjoining two columns labelled a and γ to the resulting matrix such that the column labelled a is zero everywhere except in the last row where it takes the value 1, and γ is sum of the two column vectors corresponding to the elements a and e. The vector matroid of the matrix A e X is denoted by M e X . The transition from M to M e X is called an es-splitting operation. We call the matroid M e X as es-splitting matroid.
The following proposition characterizes the circuits of the matroid M e X in terms of the circuits of the matroid M .

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On n-connected minors of the es-splitting binary matroids | P. P. Malavadkar et al. C 0 = {C ∈ C | C contains an even number of elements of X }; and each of C 1 and C 2 contains an odd number of elements 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 elements of X}; ∈ C and C contains an odd number of elements of X } ∪ {(C \ e) ∪ {γ} | C ∈ C, e ∈ C and C contains an odd number of elements of X } ∪ {(C \ e) ∪ {a, γ} | C ∈ C, e ∈ C and C \ e contains an odd number of elements of X }.
Throughout this paper we assume that M is a loopless and coloopless binary matroid, X ⊂ E(M ) and M e X is the es-splitting matroid of M . We denote by C OX the set of all circuits of a matroid M each of which contains an odd number of elements of the set X. The members of the set C OX are called OX-circuits. On the other hand, C EX denotes the set of all circuits of a matroid M each of which contains an even number of elements of the set X. The members of the set C EX are called EX-circuits.
It is intersting to observe that M e X \ γ and M e X \ {a, γ} are isomorphic with element splitting matroid and splitting matroid of M , respectively. The main theorems of this paper, Theorem 3.1 and Theorem 3.2 are motivated by a series of earlier work on splitting operation, element splitting operation and es-splitting operation [1,2,4,7,8,10,11,12,15,17].
The following result characterizes the rank function of the matroid M e X in terms of the rank function of the matroid M [4]. = r(A); otherwise.
Using Lemma 1.1, one can obtain the following corollary. We recall that matroid M is connected if and only if for every pair of distinct elements of E(M ), there is a circuit containing both. The concept of n-connected matroids was introduced by W. T. Tutte [14]. If k is positive integer, the matroid M is k-separated if there is a subset The following result from [9] provides a necessary condition for a matroid to be n-connected.
If M is a n-connected matroid and |E(M )| ≥ 2(n − 1) then all circuits and all cocircuits of M have at least n elements.
Let M be an n-connected binary matroid and X ⊂ E(M ). Note that if |X| < n then X ∪ {a} will be a cocircuit of M e X . Further, if |X ∪ {a}| < n then, by Lemma 1.2, M e X is not n-connected. Azanchiler [1] proved that es-splitting operation on a connected binary matroid yields a connected binary matroid. In fact, he proved the following theorem. In the following result Dhotre, Malavadkar and Shikare [4], provided a sufficient condition for the es-splitting operation to yield a 3-connected binary matroid from a 3-connected binary matroid.
Suppose that M has an OX-circuit not containing e. Then M e X is a 3-connected binary matroid.
In perticular, when X = {x, y} the es-spliting maroid is denoted by M e x,y . As a consequence of the above result, Dhotre, Malavadkar and Shikare [4] obtained a splitting lemma for es-splitting matroid M e x,y . Corollary 1.2. (Splitting Lemma). If M is a 3-connected binary matroid then, M e x,y is a 3connected binary matroid for any pair {x, y} of elements of E(M ).

3-Connected
Now one of the following two cases concerning a and γ occurs. In the following lemma, we provide a sufficient condition for a 3-connected binary matroid M so that M e X \ γ is a 3-connected minor of the es-splitting matroid M e X . We conclude that M e X \ γ is a 3-connected binary matroid.

n-Connected
Minors of the es-splitting Matroids.
In this section, we provide a sufficient condition for an n-connected binary matroid M (n ≥ 4) of rank r, where M e X \ e and M e X \ γ are n-connected minors of rank r + 1 of the es-splitting matroid M e X .
Let M be an n-connected binary matroid (n ≥ 4), X ⊆ E(M ) and e ∈ X. Suppose that M has an OX-circuit not containing e. Then, by Theorem 1.2, the binary matroid M e X is 3-connected. Note that the matroid M e X contains a triangle = {a, e, γ}. Hence, by Proposition 1.2, M e X is not 4-connected. We observe that for any x ∈ E(M e X ), M e X /x contains a 2-circuit or a triangle and therefore it is not 4-connected. Further, for any x ∈ (E(M e X ) − ), the minor M e X \ x contains the triangle and therefore, it is not 4-connected. Thus, the possible 4-connected minors of M e X are M e X \ e and M e X \ γ. In the following theorem, we give a sufficient condition for an n-connected binary matroid M where M e X \ e is an n-connected minor of M e X .  This leads to a k-separation of M , a contradiction. Thus, M e X has no k-separation. We conclude that M e X \ e is k + 1-connected. We conclude that, by principle of mathematical induction, the result is true for all n ≥ 4.
In the following theorem, we give a sufficient condition for an n-connected binary matroid M so that M e X \ γ is an n-connected minor of M e X .
Theorem 3.2. Let M be an n-connected binary matroid with n ≥ 4, |E(M )| ≥ 2(n − 1) and let X ⊂ E(M ), where |X| ≥ n. Suppose that for any (n − 2)-element subset S of E(M ) there is an The proof follows by the arguments similar to one as given for the proof of Theorem 3.1. Thus, we proved that given an n-connected binary matroid M of rank r, M e X \ e and M e X \ γ are the n-connected minors of rank (r + 1) of the es-splitting matroid M e X . In other words, we provide a procedure to obtain n-connected matroids of rank (r + 1) from an n-connected matroid of rank r. The matroids also have the property that each of them has exactly one additional element than M . We illustrate Theorems 3.1 and 3.2 with the help of the following example.

Acknowledgement
The authors are thankful to the anonymous referee for providing valuable suggestions which have helped to improve the presentation of the paper.