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ISRN Discrete Mathematics
Volume 2011 (2011), Article ID 629707, 8 pages
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.
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.
Slater  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 be a graph and be the set of vertices of . Let be the graph obtained from by adding a new vertex adjacent to vertices of . The graph is said to be obtained from by point-addition operation. Letting , for convenience, we denote the graph by . Thus, and .
Point-addition operation has several applications in graph theory. For example, Slater classified -connected graphs using point-addition operation along with some other operations .
If , then the new vertex can be joined to at most vertices of the graph. That means, we can add at most edges in the original graph.
Definition 1.1. Let be a binary matroid of rank on a set . Let be the matrix obtained from by the following way.(1)Adjoin columns to with labels say . Let the resulting matrix be denoted by .(2)Adjoin a new row to with entries zero except in the columns corresponding to , where it takes the value 1.
Let be the vector matroid of the matrix . We say that is obtained from by the point-addition operation. We call the matroid point-addition matroid or -extension of . Let us denote by , the set of columns which are adjoined to in the first step. That is, . Then, second step consists of splitting the matroid with respect to the set (see [3, 4]).
In fact, the matroid is obtained by elements addition and generalized splitting operation . As an immediate consequence of the definition, we have the following result.
Let and be two vertices of . Then, the addition of an edge , results in the smallest supergraph of containing edge .
Proposition 1.2. Let be a cycle matroid of rank . Let be the graph obtained from by adding adjacent edges to . Let . Then, the point-addition matroid is graphic and .
Proof. Let be representation matrix of over . Let the matrix be obtained from by adding column vectors say, . Suppose that is obtained from by adding a new row where entries are zero, except in the columns corresponding to , where it takes the value 1. Thus is a binary matroid with ground set . Since are adjacent edges in , the splitting of with respect to is graphic (see ), and we have , where is the graph obtained from by splitting operation with respect to . It follows that .
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 .
2. -Extension of a Binary Matroid
If a matroid is obtained from a matroid by adding a nonempty subset of , then is called an extension of . In particular, if , then is a single-element extension of (see ). Another term, that is sometimes used instead of single-element extension, is addition (see ).
Now we consider a special case of the operation that is introduced in the first section.
Definition 2.1. Let be a binary matroid of rank on a set , and let be the standard representation of over . Let be a base of , and let be a subset of . We obtain the matrix by the following way.(1)Obtain a matrix from by adjoining columns say to , parallel to , respectively.(2)Split the matrix with respect to the set , where . Denote the resulting matrix by .
Let be the vector matroid of the matrix . We say that is -extension of . Note that is a binary matroid with ground set , where , and . The transition from to is called -extension operation on . In particular, if , it is called --extension operation, and, for , we call it single--extension operation.
The next example illustrates this construction for the dual of Fano matroid.
Example 2.2. Let be the dual of the Fano matroid , and let be the ground set of . The matrix that represents over is given by.
Consider the set contained in the base of . Then, the corresponding matrix is given by
The vector matroid of is the matroid .
Corollary 2.3. Let be a binary matroid on . Let be a subset of a base of , and be the -extension of on the set . Then, , that is, is an extension of .
Corollary 2.4. Let and be the rank functions of the matroids and , respectively. Then .
With the help of Lemma 2.5, we characterize the circuits of the matroid .
Lemma 2.5. (1) Every circuit of is a circuit of .
(2) Every circuit of contains at least one element of .
(3) Every circuit of contains even number of elements of .
The proof follows from the construction of the matrix .
Remark 2.6. Let be a single--extension of (i.e., ). Then every circuit of is a circuit of and vice versa. In fact, the added element is a coloop in the resulting matroid.
Theorem 2.7. Let be a binary matroid on with representation matrix over and be a subset of a base of . Then, a subset of is a circuit of if and only if one of the following conditions hold:(1), where and for ,(2), where , is an even integer and is such that is a circuit in , where .
Proof. If , then, by Definition 2.1 of , is a circuit of . Now, let be as stated in (2). If , then , and is a circuit of . Suppose that , and is a circuit of . Then clearly, is a circuit of .
Conversely, let be a circuit of , we have two cases:(I). Then . Thus, is a circuit in and the condition (2) in the result holds.(II)Let , and suppose that . We have two subcases:(i). Then, and . Thus, and condition (1) in the result holds.(ii). Take . Then is a circuit of and is a circuit of . Thus, , and the condition (2) in the result holds.
We characterize the independent sets of in terms of independent sets of . Firstly, we have the following lemma.
Lemma 2.8. (1) Every independent set of is independent in .
(2) Every subset of is independent in .
The proof is straightforward.
Remark 2.9. Let be a single--extension of . Then, every independent set in is also independent in and vice versa.
Theorem 2.10. Let be a binary matroid on and be the -extension matroid of with respect to . Let be a collection of independent sets of . Then, a subset of is an independent set of if and only if one of the following conditions hold:(1), where and .(2), where , and contains no circuit of .
Proof. If , then clearly for is an independent set in . Now, suppose that contains no circuit of . On the contrary, suppose that contains a circuit say of . Then and . But is a circuit of and is contained in , a contradiction.
Conversely, let be an independent set in and . We have two cases.(I)Let . Then and is independent in .(II)Let , and let . Then and .
We prove that is an independent set in . On the contrary, suppose that contains a circuit of , say , then and gives a contradiction.
Letting , we have . We claim that does not contain any circuit of . If contains a circuit of , say , then . Further, , and thus leads to a contradiction as is a circuit of . This completes the proof of the theorem.
Corollary 2.11. Let and denote the collection of independent sets of and , respectively. If , then .
Corollary 2.12. A subset of is independent in if and only if for is independent in .
Corollary 2.13. Let and be the rank functions of and , respectively. Then for .
In the next theorem, we characterize the bases of the matroid in terms of the bases of .
Lemma 2.14. Let , then is an independent set in if and only if is an independent set in .
The proof is straightforward.
Corollary 2.15. Let be a subset of . Then where and are rank functions of and , respectively.
Theorem 2.16. A subset of is a base for if and only if , where and contains no circuit of .
Proof. Let be a base for . Then is an independent set in , and so is independent in . Let . Then, by Theorem 2.10, , where and contains no circuit of and hence is independent in . Moreover, .
We conclude that is a base for .
Conversely, let be a base for . Firstly, we show that . On the contrary, suppose that . Then and is independent in . So by Lemma 2.14, is independent in . Also by Corollary 2.15, This shows that and ; a contradiction.
Now, let . Then is independent in as well as in . It can be extended to form the base of . Let be such that is a base for . Then . We claim that . Now, Since , we conclude that , that is, . Finally, we show that contains no circuit of . On the contrary, suppose that contains a circuit, say of . Then and . Thus, leads to a contradiction. This completes the proof of theorem.
Corollary 2.17. Every base of contains at least one element of .
3. Connectivity of
Let be a binary matroid on a set and be the representation matrix of over . If is bridgeless, then -extension of with respect to a singleton subset of yields a disconnected matroid.
Lemma 3.1. Let be a coloop in a matroid and . Then is a coloop in .
The proof is straightforward.
Corollary 3.2. Suppose that no element of is a coloop of . Then has no coloops.
Theorem 3.3. Let . If is connected matroid, then, so is .
Proof. Assume that is connected. We show that for every pair of elements there is a circuit of containing and . We have three cases.(1)Let . By hypothesis, is connected. So there is a circuit of say , containing and . Since is a circuit in , we are through.(2)Let and let and . Then the 4-circuit in contains and .(3)Let and . By assumption . So there is an element say . Let be a circuit of containing and . By Lemma 2.5, contains at least one element of , say . Now , and is connected, so there is a circuit of , say which contains and . Thus and . Then there is a circuit in , say , such that and . This completes the proof of the theorem.
Remark 3.4. Converse of the above theorem is not true.
Theorem 3.5. Let be a 3--extension matroid of and . If is a 3-connected matroid on , then is 3-connected.
Proof. On the contrary, suppose that is not 3-connected, then has a 1-separated or 2-separated partition. Let be a 2-separated partition of . That means, and We consider three cases.(i)Let and . By Lemma 2.8, is independent in , so Also, by Lemma 2.14, Thus, This is a contradiction to (*).(ii)Let and .By Lemma 2.14, . So gives a contradiction to (*).(iii)Let and , where , and . Then and . Moreover, Thus, , and we conclude that is a 1-separated partition for .This is a contradiction to the fact that is 3-connected. By the same argument, we can show that does not have 1-separated partition.
In the last theorem, the condition that is necessary. Consider the following example.
Example 3.6. is a 3-connected matroid. Let a representation matrix of be Let and . Then By row operations on , we can show that By , is not 3-connected.
If , then has a coloop, and it is not 3-connected.
In general, we state the following result whose proof is immediate.
Corollary 3.7. Let be a -connected binary matroid and . Then is not -connected.
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