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ISRN Discrete Mathematics
Volume 2011 (2011), Article ID 629707, 8 pages
http://dx.doi.org/10.5402/2011/629707
Research Article

Γ-Extension of Binary Matroids

Department of Mathematics, Faculty of Sciences, University of Urmia, P.O. Box 57135, Urmia, Iran

Received 13 August 2011; Accepted 22 September 2011

Academic Editors: Y. Hou and T. Prellberg

Copyright © 2011 Habib Azanchiler. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We extend the notion of a point-addition operation from graphs to binary matroids. This operation can be expressed in terms of element-addition operation and splitting operation. We consider a special case of this construction and study its properties. We call the resulting matroid of this special case a Γ-extension of the given matroid. We characterize circuits and bases of the resulting matroids and explore the effect of this operation on the connectivity of the matroids.

1. Introduction

Slater [1] defined few operations for graphs which preserve connectedness of graphs. One such operation is a point-addition (vertex-addition) operation. This operation is defined in the following way. Let 𝐺 be a graph and 𝑉(𝐺) be the set of vertices of 𝐺. Let 𝐻 be the graph obtained from 𝐺 by adding a new vertex 𝑣 adjacent to vertices 𝑣1,𝑣2,,𝑣𝑛 of 𝐺. The graph 𝐻 is said to be obtained from 𝐺 by point-addition operation. Letting 𝑋={𝑣1,𝑣2,,𝑣𝑛}, for convenience, we denote the graph 𝐻 by 𝐺𝑋. Thus, 𝑉(𝐺𝑋)=𝑉(𝐺){𝑣} and 𝐸(𝐺𝑋)=𝐸(𝐺){𝑣𝑣1,𝑣𝑣2,,𝑣𝑣𝑛}.

Point-addition operation has several applications in graph theory. For example, Slater classified 𝑛-connected graphs using point-addition operation along with some other operations [2].

If |𝑉(𝐺)|=𝑛, then the new vertex 𝑣 can be joined to at most 𝑛 vertices of the graph. That means, we can add at most 𝑛 edges in the original graph.

Definition 1.1. Let 𝑀=𝑀[𝐴] be a binary matroid of rank 𝑟 on a set 𝑆. Let 𝐴 be the matrix obtained from 𝐴 by the following way.(1)Adjoin 𝑘 columns to 𝐴(𝑘𝑟) with labels say 𝛾1,𝛾2,,𝛾𝑘. Let the resulting matrix be denoted by 𝐴.(2)Adjoin a new row to 𝐴 with entries zero except in the columns corresponding to 𝛾1,𝛾2,,𝛾𝑘, where it takes the value 1.

Let 𝑀 be the vector matroid of the matrix 𝐴. We say that 𝑀 is obtained from 𝑀 by the point-addition operation. We call the matroid 𝑀 point-addition matroid or Γ-extension of 𝑀. Let us denote by Γ, the set of columns 𝛾1,𝛾2,,𝛾𝑘 which are adjoined to 𝐴 in the first step. That is, Γ={𝛾1,𝛾2,,𝛾𝑘}. Then, second step consists of splitting the matroid 𝑀[𝐴] with respect to the set Γ (see [3, 4]).

In fact, the matroid 𝑀 is obtained by elements addition and generalized splitting operation [5]. As an immediate consequence of the definition, we have the following result.

Let 𝑣𝑖 and 𝑣𝑗 be two vertices of 𝐺. Then, the addition of an edge 𝑣𝑖𝑣𝑗, results in the smallest supergraph of 𝐺 containing edge 𝑣𝑖𝑣𝑗.

Proposition 1.2. Let 𝑀=𝑀(𝐺) be a cycle matroid of rank 𝑟. Let 𝐺 be the graph obtained from 𝐺 by adding adjacent edges 𝛾1,𝛾2,,𝛾𝑛(𝑛𝑟) to 𝐺. Let Γ={𝛾1,𝛾2,,𝛾𝑛}. Then, the point-addition matroid 𝑀 is graphic and 𝑀=𝑀(𝐺Γ).

Proof. Let 𝐴 be representation matrix of 𝑀 over 𝐺𝐹(2). Let the matrix 𝐴 be obtained from 𝐴 by adding column vectors say, 𝛾1,𝛾2,,𝛾𝑛. Suppose that 𝐴 is obtained from 𝐴 by adding a new row where entries are zero, except in the columns corresponding to 𝛾1,𝛾2,,𝛾𝑛, where it takes the value 1. Thus 𝑀=𝑀[𝐴] is a binary matroid with ground set 𝐸(𝐺)Γ. Since 𝛾1,𝛾2,,𝛾𝑛 are adjacent edges in 𝐺, the splitting of 𝑀(𝐺) with respect to Γ is graphic (see [5]), and we have 𝑀(𝐺)Γ=𝑀(𝐺Γ), where 𝐺Γ is the graph obtained from 𝐺 by splitting operation with respect to Γ. It follows that 𝑀=[𝑀(𝐺)]=𝑀(𝐺Γ).

We assume that the reader is familiar with elementary notions in matroid theory, including minors, binary, and connectivity. For an excellent introduction to the subject, read Oxley [6].

2. Γ-Extension of a Binary Matroid

If a matroid 𝑀 is obtained from a matroid 𝑁 by adding a nonempty subset 𝑇 of 𝑆(𝑁), then 𝑁 is called an extension of 𝑀. In particular, if |𝑇|=1, then 𝑁 is a single-element extension of 𝑀 (see [6]). Another term, that is sometimes used instead of single-element extension, is addition (see [7]).

Now we consider a special case of the operation that is introduced in the first section.

Definition 2.1. Let 𝑀=𝑀[𝐴] be a binary matroid of rank 𝑟 on a set 𝑆, and let 𝐴=[𝐼𝑟𝐽] be the standard representation of 𝑀 over 𝐺𝐹(2). Let 𝐵 be a base of 𝑀, and let 𝑋={𝑒𝑖1,𝑒𝑖2,,𝑒𝑖𝑚} be a subset of 𝐵. We obtain the matrix 𝐴𝑋 by the following way.(1)Obtain a matrix 𝐴1 from 𝐴 by adjoining 𝑚(𝑚𝑟) columns say 𝛾𝑖1,𝛾𝑖2,,𝛾𝑖𝑚 to 𝐴, parallel to 𝑒𝑖1,𝑒𝑖2,,𝑒𝑖𝑚, respectively.(2)Split the matrix 𝐴1 with respect to the set Γ, where Γ={𝛾𝑖1,𝛾𝑖2,,𝛾𝑖𝑚}. Denote the resulting matrix by 𝐴𝑋.

Let 𝑀𝑋 be the vector matroid of the matrix 𝐴𝑋. We say that 𝑀𝑋 is Γ-extension of 𝑀. Note that 𝑀𝑋 is a binary matroid with ground set 𝑆Γ, where 𝑆Γ=𝜙, and |𝑋|=|Γ|. The transition from 𝑀 to 𝑀𝑋 is called Γ-extension operation on 𝑀. In particular, if |𝑋|=, it is called -Γ-extension operation, and, for |𝑋|=1, we call it single-Γ-extension operation.

The next example illustrates this construction for the dual of Fano matroid.

Example 2.2. Let 𝑀=𝐹7 be the dual of the Fano matroid 𝐹7, and let 𝑆={1,2,3,4,5,6,7} be the ground set of 𝑀. The matrix 𝐴 that represents 𝑀 over 𝐺𝐹(2) is given by. 1234567𝐴=1000110010001100101010001111.(2.1)

Consider the set 𝑋={1,3,4} contained in the base of 𝑀. Then, the corresponding matrix 𝐴𝑋 is given by 1234567𝛾1𝛾3𝛾4𝐴𝑋=10001101000100011000001010101000011110010000000111.(2.2)

The vector matroid of 𝐴𝑋 is the matroid (𝐹7)𝑋.

Corollary 2.3. Let 𝑀=𝑀[𝐴] be a binary matroid on 𝑆. Let 𝑋 be a subset of a base of 𝑀, and 𝑀𝑋 be the Γ-extension of 𝑀 on the set 𝑆Γ. Then, 𝑀𝑋Γ=𝑀, that is, 𝑀𝑋 is an extension of 𝑀.

Corollary 2.4. Let 𝑟 and 𝑟 be the rank functions of the matroids 𝑀 and 𝑀𝑋, respectively. Then 𝑟(𝑀𝑋)=𝑟(𝑀)+1.

With the help of Lemma 2.5, we characterize the circuits of the matroid 𝑀𝑋.

Lemma 2.5. (1) Every circuit of 𝑀 is a circuit of 𝑀𝑋.
(2) Every circuit of 𝑀𝑋 contains at least one element of 𝑆.
(3) Every circuit of 𝑀𝑋 contains even number of elements of Γ.

The proof follows from the construction of the matrix 𝐴𝑋.

Remark 2.6. Let 𝑀𝑋 be a single-Γ-extension of 𝑀 (i.e., |𝑋|=1). Then every circuit of 𝑀𝑋 is a circuit of 𝑀 and vice versa. In fact, the added element 𝛾 is a coloop in the resulting matroid.

Theorem 2.7. Let 𝑀 be a binary matroid on 𝑆 with representation matrix 𝐴=[𝐼𝑟𝐸] over 𝐺𝐹(2) and 𝑋 be a subset of a base of 𝑀. Then, a subset 𝑍 of 𝑆Γ is a circuit of 𝑀𝑋 if and only if one of the following conditions hold:(1)𝑍={𝑒𝑖,𝑒𝑗,𝛾𝑖,𝛾𝑗}, where 𝑖𝑗,𝛾𝑖,𝛾𝑗Γ and 𝑒𝑖,𝑒𝑗𝑋 for 1𝑖,𝑗𝑟,(2)𝑍=𝐽𝐷, where 𝐽Γ, |𝐽| is an even integer and 𝜙𝐷𝑆 is such that 𝐷𝑋𝐽 is a circuit in 𝑀, where 𝑋𝐽={𝑒𝑖𝑋𝛾𝑖𝐽}.

Proof. If 𝑍={𝑒𝑖,𝑒𝑗,𝛾𝑖,𝛾𝑗}, then, by Definition 2.1 of 𝐴𝑋, 𝑍 is a circuit of 𝑀𝑋. Now, let 𝑍=𝐽𝐷 be as stated in (2). If |𝐽|=0, then 𝐽=𝜙,𝑋𝐽=𝜙, and 𝑍=𝐷 is a circuit of 𝑀. Suppose that 𝐽Γ,𝐽𝜙, and 𝑋𝐽𝐷 is a circuit of 𝑀. Then clearly, 𝑍=𝐷𝐽 is a circuit of 𝑀𝑋.
Conversely, let 𝑍𝑆Γ be a circuit of 𝑀𝑋, we have two cases:(I)𝑍Γ=𝜙. Then 𝐽=𝜙,𝑋𝐽=𝜙. Thus, 𝑍=𝐷 is a circuit in 𝑀 and the condition (2) in the result holds.(II)Let 𝑍Γ𝜙, and suppose that 𝑍Γ=𝐽. We have two subcases:(i)𝑍𝑋𝜙. Then, 𝑍𝑋={𝑒𝑖,𝑒𝑗}=𝐷=𝑋𝐽 and 𝐽={𝛾𝑖,𝛾𝑗}. Thus, 𝑍={𝑒𝑖,𝑒𝑗,𝛾𝑖,𝛾𝑗} and condition (1) in the result holds.(ii)𝑍𝑋=𝜙. Take 𝑍𝑆=𝐷. Then 𝐷𝑋𝐽 is a circuit of 𝑀 and 𝐷𝐽 is a circuit of 𝑀𝑋. Thus, 𝑍=𝐷𝐽, and the condition (2) in the result holds.

We characterize the independent sets of 𝑀𝑋 in terms of independent sets of 𝑀. Firstly, we have the following lemma.

Lemma 2.8. (1) Every independent set of 𝑀 is independent in 𝑀𝑋.
(2) Every subset of Γ is independent in 𝑀𝑋.

The proof is straightforward.

Remark 2.9. Let 𝑀𝑋 be a single-Γ-extension of 𝑀. Then, every independent set in 𝑀𝑋 is also independent in 𝑀 and vice versa.

Theorem 2.10. Let 𝑀 be a binary matroid on 𝑆 and 𝑀𝑋 be the Γ-extension matroid of 𝑀 with respect to 𝑋. Let be a collection of independent sets of 𝑀. Then, a subset 𝐼 of 𝑆Γ is an independent set of 𝑀𝑋 if and only if one of the following conditions hold:(1)𝐼=𝐼1{𝛾}, where 𝛾Γ and 𝐼1.(2)𝐼=𝐼1𝐽, where 𝐽Γ, 𝐼1 and 𝐼1𝑋𝐽 contains no circuit of 𝑀.

Proof. If 𝐼1, then clearly 𝐼1{𝛾} for 𝛾Γ is an independent set in 𝑀𝑋. Now, suppose that 𝐼1𝑋𝐽 contains no circuit of 𝑀. On the contrary, suppose that 𝐼1𝐽 contains a circuit say 𝐶 of 𝑀𝑋. Then 𝐶𝐽𝐼1 and (𝐶𝐽)𝑋𝐽𝐼1𝑋𝐽. But (𝐶𝐽)𝑋𝐽 is a circuit of 𝑀 and is contained in 𝐼1𝑋𝐽, a contradiction.
Conversely, let 𝐼 be an independent set in 𝑀𝑋 and 𝐼𝑆Γ. We have two cases.(I)Let 𝐼Γ=𝜙. Then 𝐼𝑆 and 𝐼 is independent in 𝑀.(II)Let 𝐼Γ𝜙, and let 𝐼Γ=𝐽. Then 𝐽Γ and 𝐽𝐼.
We prove that 𝐼𝐽𝑆 is an independent set in 𝑀. On the contrary, suppose that 𝐼𝐽 contains a circuit of 𝑀, say 𝐶, then 𝐶𝐼𝐽 and 𝐶𝐼 gives a contradiction.
Letting 𝐼𝐽=𝐼1, we have 𝐼=𝐼1𝐽. We claim that 𝐼1𝑋𝐽 does not contain any circuit of 𝑀. If 𝐼1𝑋𝐽 contains a circuit of 𝑀, say 𝐶, then 𝐶𝑋𝐽𝐼1. Further, (𝐶𝑋𝐽)𝐽𝐼1𝐽, and thus (𝐶𝑋𝐽)𝐽𝐼 leads to a contradiction as (𝐶𝑋𝐽)𝐽 is a circuit of 𝑀𝑋. This completes the proof of the theorem.

Corollary 2.11. Let and 𝑋 denote the collection of independent sets of 𝑀 and 𝑀𝑋, respectively. If |𝑋|=1, then 𝑋={𝐼{𝛾}𝐼,𝛾Γ}.

Corollary 2.12. A subset 𝐼 of 𝑆 is independent in 𝑀 if and only if 𝐼{𝛾} for 𝛾Γ is independent in 𝑀𝑋.

Corollary 2.13. Let 𝑟 and 𝑟 be the rank functions of 𝑀 and 𝑀𝑋, respectively. Then 𝑟(𝑍{𝛾})=𝑟(𝑍)+1 for 𝑍𝑆.

In the next theorem, we characterize the bases of the matroid 𝑀𝑋 in terms of the bases of 𝑀.

Lemma 2.14. Let 𝑍𝑆, then 𝑍 is an independent set in 𝑀𝑋 if and only if 𝑍 is an independent set in 𝑀.

The proof is straightforward.

Corollary 2.15. Let 𝑍 be a subset of 𝑆. Then 𝑟(𝑍)=𝑟(𝑍) where 𝑟 and 𝑟 are rank functions of 𝑀𝑋 and 𝑀, respectively.

Theorem 2.16. A subset 𝐵 of 𝑆Γ is a base for 𝑀𝑋 if and only if 𝐵=(𝐵𝐷)𝐽, where 𝜙𝐽Γ,𝐷𝐵,|𝐷|=|𝐽|1 and (𝐵𝐷)𝑋𝐽 contains no circuit of 𝑀.

Proof. Let 𝐵 be a base for 𝑀. Then 𝐵 is an independent set in 𝑀, and so 𝐵𝐷 is independent in 𝑀. Let 𝐵𝐷=𝐼1. Then, by Theorem 2.10, 𝐼1𝐽, where 𝐽Γ,|𝐷|=|𝐽|1 and (𝐵𝐷)𝑋𝐽 contains no circuit of 𝑀 and hence is independent in 𝑀𝑋. Moreover, 𝑟((𝐵𝐷)𝐽)=|(𝐵𝐷)𝐽|=|𝐵||𝐷|+|𝐽|=|𝐵|+1=𝑟(𝑀𝑋).
We conclude that 𝐵=(𝐵𝐷)𝐽 is a base for 𝑀𝑋.
Conversely, let 𝐵𝑆Γ be a base for 𝑀𝑋. Firstly, we show that 𝐵Γ𝜙. On the contrary, suppose that 𝐵Γ=𝜙. Then 𝐵𝑆 and is independent in 𝑀𝑋. So by Lemma 2.14, 𝐵 is independent in 𝑀. Also by Corollary 2.15, 𝑟𝐵||𝐵=𝑟(𝐵)=||=𝑟(𝑀)+1.(2.3) This shows that 𝐵𝑆 and 𝑟(𝐵)>𝑟(𝑀); a contradiction.
Now, let 𝐵Γ=𝐽. Then 𝐵𝐽𝑆 is independent in 𝑀𝑋 as well as in 𝑀. It can be extended to form the base of 𝑀. Let 𝐷𝑆(𝐵𝐽) be such that (𝐵𝐽)𝐷=𝐵 is a base for 𝑀. Then 𝐵=(𝐵𝐷)𝐽. We claim that |𝐷|=|𝐽|1. Now, ||𝐵||=||𝐵||=||𝐵𝐽𝐷||||𝐽||+||𝐷||||𝐽||+||𝐷||=𝑟(𝑀)+1.(2.4) Since |𝐵|=𝑟(𝑀), we conclude that 1|𝐽|+|𝐷|=0, that is, |𝐷|=|𝐽|1. Finally, we show that (𝐵𝐷)𝑋𝐽 contains no circuit of 𝑀. On the contrary, suppose that (𝐵𝐷)𝑋𝐽 contains a circuit, say 𝐶 of 𝑀. Then 𝐶𝑋𝐽𝐵𝐷 and (𝐶𝑋𝐽)𝐽(𝐵𝐷)𝐽. Thus, (𝐶𝑋𝐽)𝐽𝐵 leads to a contradiction. This completes the proof of theorem.

Corollary 2.17. Every base of 𝑀𝑋 contains at least one element of Γ.

3. Connectivity of 𝑀𝑋

Let 𝑀 be a binary matroid on a set 𝑆 and 𝐴 be the representation matrix of 𝑀 over 𝐺𝐹(2). If 𝑀 is bridgeless, then Γ-extension of 𝑀 with respect to a singleton subset 𝑋 of 𝐵 yields a disconnected matroid.

Lemma 3.1. Let 𝑒𝑖𝑆 be a coloop in a matroid 𝑀 and 𝑒𝑖𝑋. Then 𝑒𝑖 is a coloop in 𝑀𝑋.

The proof is straightforward.

Corollary 3.2. Suppose that no element of 𝑆𝑋 is a coloop of 𝑀. Then 𝑀𝑋 has no coloops.

Theorem 3.3. Let |𝑋|2. If 𝑀 is connected matroid, then, so is 𝑀𝑋.

Proof. Assume that 𝑀 is connected. We show that for every pair of elements 𝑢,𝑣𝑆Γ there is a circuit of 𝑀𝑋 containing 𝑢 and 𝑣. We have three cases.(1)Let 𝑢,𝑣𝑆. By hypothesis, 𝑀 is connected. So there is a circuit of 𝑀 say 𝐶, containing 𝑢 and 𝑣. Since 𝐶 is a circuit in 𝑀𝑋, we are through.(2)Let 𝑢,𝑣Γ and let 𝑢=𝛾𝑖 and 𝑣=𝛾𝑗. Then the 4-circuit {𝑒𝑖,𝑒𝑗,𝛾𝑖,𝛾𝑗} in 𝑀𝑋 contains 𝑢 and 𝑣.(3)Let 𝑢𝑆 and 𝑣Γ. By assumption |𝑋|2. So there is an element say 𝑤Γ. Let 𝐶1 be a circuit of 𝑀𝑋 containing 𝑣 and 𝑤. By Lemma 2.5, 𝐶1 contains at least one element of 𝑆, say 𝑒. Now 𝑢,𝑒𝑆, and 𝑀 is connected, so there is a circuit of 𝑀, say 𝐶2 which contains 𝑢 and 𝑒. Thus 𝐶1𝐶2𝜙 and 𝑢𝐶1,𝑣𝐶2. Then there is a circuit in 𝑀𝑋, say 𝐶3, such that 𝐶3𝐶1𝐶2 and 𝑢,𝑣𝐶3. This completes the proof of the theorem.

Remark 3.4. Converse of the above theorem is not true.

Theorem 3.5. Let 𝑀𝑋 be a 3-Γ-extension matroid of 𝑀 and Γ={𝛾𝑖,𝛾𝑗,𝛾𝑘}. If 𝑀 is a 3-connected matroid on 𝑆, then 𝑀𝑋 is 3-connected.

Proof. On the contrary, suppose that 𝑀𝑋 is not 3-connected, then 𝑀𝑋 has a 1-separated or 2-separated partition. Let (𝐴,𝐵) be a 2-separated partition of 𝑆Γ. That means, min{|𝐴|,|𝐵|}2 and 𝑟(𝐴)+𝑟(𝐵)𝑟𝑀𝑋1.() We consider three cases.(i)Let 𝐴=Γ and 𝐵=𝑆. By Lemma 2.8, 𝐴 is independent in 𝑀𝑋, so 𝑟||𝐴||=||Γ||(𝐴)==3.(3.1) Also, by Lemma 2.14, 𝑟(𝐵)=𝑟(𝑆)=𝑟(𝑆)=𝑟(𝑀).(3.2) Thus, 𝑟(𝐴)+𝑟(𝐵)𝑟𝑀𝑋=3+𝑟(𝑀)𝑟(𝑀)1=2.(3.3) This is a contradiction to (*).(ii)Let 𝐴={𝛾𝑖,𝛾𝑗} and 𝐵=𝑆{𝛾𝑘}.By Lemma 2.14, 𝑟(𝐵)=𝑟(𝑀)+1. So 𝑟(𝐴)+𝑟(𝐵)𝑟𝑀𝑋=2+𝑟(𝑀)+1𝑟(𝑀)1=2(3.4)gives a contradiction to (*).(iii)Let 𝐴=𝑆1{𝛾𝑖} and 𝐵=𝑆2{𝛾𝑗,𝛾𝑘}, where 𝑆1,𝑆2𝑆, and 𝑆1𝜙,𝑆2𝜙. Then min{|𝑆1|,|𝑆2|}1 and 𝑟(𝑆1)+𝑟(𝑆2)𝑟(𝑀)𝑟(𝐴)1+𝑟(𝐵)1𝑟(𝑀𝑋)+10. Moreover, 𝑟𝑆1𝑆+𝑟2𝑟(𝑀).(3.5) Thus, 𝑟(𝑆1)+𝑟(𝑆2)𝑟(𝑀)=0, and we conclude that (𝑆1,𝑆2) is a 1-separated partition for 𝑀.This is a contradiction to the fact that 𝑀 is 3-connected. By the same argument, we can show that 𝑀𝑋 does not have 1-separated partition.

In the last theorem, the condition that |𝑋|=3 is necessary. Consider the following example.

Example 3.6. 𝑀(𝐾4) is a 3-connected matroid. Let a representation matrix of 𝑀(𝐾4) be 123456𝐴=100110010101001011.(3.6) Let 𝑋𝐵 and 𝑋={1,2},Γ={𝛾𝑖,𝛾2}. Then 123456𝛾1𝛾2𝐴𝑋=10011010010101010010110000000011.(3.7) By row operations on 𝐴𝑋, we can show that 𝐴𝑋=𝐼4||||||1101101101100001.(3.8) By [], 𝑀𝑋=𝑀[𝐴𝑋] is not 3-connected.

If |𝑋|=1, then 𝑀𝑋 has a coloop, and it is not 3-connected.

In general, we state the following result whose proof is immediate.

Corollary 3.7. Let 𝑀 be a 𝑛-connected binary matroid and |𝑋|<𝑛. Then 𝑀𝑋 is not 𝑛-connected.

References

  1. P. J. Slater, “A classification of 4-connected graphs,” Journal of Combinatorial Theory. Series B, vol. 17, pp. 281–298, 1974. View at Publisher · View at Google Scholar
  2. P. J. Slater, “Soldering and point splitting,” Journal of Combinatorial Theory. Series B, vol. 24, no. 3, pp. 338–343, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. T. T. Raghunathan, M. M. Shikare, and B. N. Waphare, “Splitting in a binary matroid,” Discrete Mathematics, vol. 184, no. 1–3, pp. 267–271, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. M. M. Shikare, “Splitting operation and connectedness in binary matroids,” Indian Journal of Pure and Applied Mathematics, vol. 31, no. 12, pp. 1691–1697, 2000. View at Zentralblatt MATH
  5. M. M. Shikare, G. Azadi, and B. N. Waphare, “Generalized splitting operation and its applications to binary matroids,” Preprint.
  6. J. G. Oxley, Matroid Theory, Oxford Science Publications, The Clarendon Press/Oxford University Press, New York, NY, USA, 1992.
  7. K. Truemper, Matroid Decomposition, Academic Press, Boston, Mass, USA, 1992.