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 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 [2].
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 [5]. 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 [5]), 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 [6].
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 [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 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.