Γ-Extension of Binary Matroids

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.


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 v 1 , v 2 , . . ., v n of G.The graph H is said to be obtained from G by point-addition operation.Letting X {v 1 , v 2 , . . ., v n }, for convenience, we denote the graph H by G X .Thus, V G X  V G ∪ {v} and E G X E G ∪ {vv 1 , vv 2 , . . ., vv n }.
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.
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 γ 1 , γ 2 , . . ., γ k which are adjoined to A in the first step.That is, Γ {γ 1 , γ 2 , . . ., γ k }.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 v i and v j be two vertices of G.Then, the addition of an edge v i v j , results in the smallest supergraph of G containing edge v i v j .
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 .

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 I r | J be the standard representation of M over GF 2 .Let B be a base of M, and let X {e i 1 , e i 2 , . . ., e i m } be a subset of B. We obtain the matrix A X by the following way.
1 Obtain a matrix A 1 from A by adjoining m m ≤ r columns say γ i 1 , γ i 2 , . . ., γ i m to A, parallel to e i 1 , e i 2 , . . ., e i m , respectively.
2 Split the matrix A 1 with respect to the set Γ, where Γ {γ i 1 , γ i 2 , . . ., γ i m }.Denote the resulting matrix by A X .
Let M X be the vector matroid of the matrix A X .We say that M X is Γ-extension of M. Note that M X is a binary matroid with ground set S ∪ Γ, where S ∩ Γ φ, and |X| |Γ|.The transition from M to M X 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 F * 7 be the dual of the Fano matroid F 7 , 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.
Consider the set X {1, 3, 4} contained in the base of M.Then, the corresponding matrix A X is given by The vector matroid of A X is the matroid F * 7 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 M X be the Γ-extension of M on the set S ∩ Γ.Then, M X \ Γ M, that is, M X is an extension of M.
Corollary 2.4.Let r and r be the rank functions of the matroids M and M X , respectively.Then r M X r M 1.
With the help of Lemma 2.5, we characterize the circuits of the matroid M X .

Lemma 2.5. (1)
Every circuit of M is a circuit of M X .
(2) Every circuit of M X contains at least one element of S.
(3) Every circuit of M X contains even number of elements of Γ.
The proof follows from the construction of the matrix A X .
Remark 2.6.Let M X be a single-Γ-extension of M i.e., |X| 1 .Then every circuit of M X 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 I r | E over GF 2 and X be a subset of a base of M.Then, a subset Z of S ∪ Γ is a circuit of M X if and only if one of the following conditions hold: 1 Z {e i , e j , γ i , γ j }, where i / j, γ i , γ j ∈ Γ and e i , e j ∈ X for 1 ≤ i, j ≤ r, Proof.If Z {e i , e j , γ i , γ j }, then, by Definition 2.1 of A X , Z is a circuit of M X .Now, let Z J ∪D be as stated in 2 .If |J| 0, then J φ, X J φ, and Z D is a circuit of M. Suppose that J ⊆ Γ, J / φ, and Conversely, let Z ⊆ S ∪ Γ be a circuit of M X , we have two cases: Then J φ, X J φ. Thus, Z D is a circuit in M and the condition 2 in the result holds.
II Let Z ∩ Γ / φ, and suppose that Z ∩ Γ J.We have two subcases: i Z ∩ X / φ.Then, Z ∩ X {e i , e j } D X J and J {γ i , γ j }.Thus, Z {e i , e j , γ i , γ j } and condition 1 in the result holds.ii Z ∩ X φ.Take Z ∩ S D. Then D ∪ X J is a circuit of M and D ∪ J is a circuit of M X .Thus, Z D ∪ J, and the condition 2 in the result holds.
We characterize the independent sets of M X in terms of independent sets of M. Firstly, we have the following lemma.

Lemma 2.8. (1) Every independent set of
(2) Every subset of Γ is independent in M X .
The proof is straightforward.
Remark 2.9.Let M X be a single-Γ-extension of M.Then, every independent set in M X is also independent in M and vice versa.
Theorem 2.10.Let M be a binary matroid on S and M X be the Γ-extension matroid of M with respect to X. Let I be a collection of independent sets of M.Then, a subset I of S ∪ Γ is an independent set of M X if and only if one of the following conditions hold: 1 I I 1 ∪ {γ}, where γ ∈ Γ and I 1 ∈ I.
2 I I 1 ∪ J, where J ⊆ Γ, I 1 ∈ I and I 1 ∪ X J contains no circuit of M.
Proof.If I 1 ∈ I, then clearly I 1 ∪ {γ} for γ ∈ Γ is an independent set in M X .Now, suppose that I 1 ∪ X J contains no circuit of M. On the contrary, suppose that Conversely, let I be an independent set in M X and I ⊆ S ∪ Γ.We have two cases.
I Let I ∩ Γ φ.Then I ⊆ S and I is independent in M.
II Let I ∩ Γ / φ, and let I ∩ Γ J. Then J ⊆ Γ and J ⊆ I.
We prove that I − J ⊆ S is an independent set in M. On the contrary, suppose that I − J contains a circuit of M, say C, then C ⊆ I − J and C ⊆ I gives a contradiction.Also, by Lemma 2.14, This is a contradiction to * .
gives a contradiction to * .
iii Let A S 1 ∪ {γ i } and B S 2 ∪ {γ j , γ k }, where S 1 , S 2 ⊆ S, and Thus, r S 1 r S 2 − r M 0, and we conclude that S 1 , S 2 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 M X does not have 1-separated partition.
In the last theorem, the condition that |X| 3 is necessary.Consider the following example.

3.7
By row operations on A X , we can show that By , M X M A X is not 3-connected.
If |X| 1, then M X 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 M X is not n-connected.

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 γ 1 , γ 2 , . . ., γ n n ≤ r to G. Let Γ {γ 1 , γ 2 , . . ., γ n }.Then, the point-addition matroid M is graphic and M M G Γ .Proof.Let A be representation matrix of M over GF 2 .Let the matrix A be obtained from A by adding column vectors say, γ 1 , γ 2 , . . ., γ n .Suppose that A is obtained from A by adding a new row where entries are zero, except in the columns corresponding to γ 1 , γ 2 , . . ., γ n , where it takes the value 1.Thus M M A is a binary matroid with ground set E G ∪ Γ.Since γ 1 , γ 2 , . . ., γ n 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 G Γ , where G Γ is the graph obtained from G by splitting operation with respect to Γ.It follows that M