Chapter 2. Splitting Operation and n-connected Matroids. 2.1 Introduction

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1 Chapter 2 Splitting Operation and n-connected Matroids The splitting operation on an n-connected binary matroid may not yield an n-connected binary matroid. In this chapter, we provide a necessary and sufficient condition under which the generalized splitting operation on an n-connected binary matroid yields an n-connected binary matroid. A characterization of n-connected matroids has also been provided. 2.1 Introduction Fleischner [13] introduced the idea of splitting a vertex of degree at least three in a connected graph and used the operation to explore several properties of Eulerian graphs. He defined the splitting operation for a graph with respect to a pair of adjacent edges as follows: Let G be a connected graph and let v be a vertex of degree at least three in G. If x = vv 1 and y = vv 2 are two edges incident at v, then splitting

2 2.1 Introduction 20 away the pair x, y from v results in a new graph G x,y obtained from G by deleting the edges x and y, and adding a new vertex v x,y adjacent to v 1 and v 2. The transition from G to G x,y is called the splitting operation on G. For practical purposes, we denote the new edges v x,y v 1 and v x,y v 2 in G x,y by x and y, respectively. We illustrate this construction by the following figure. v 1 v 1 x x y v y v v x,y v 2 v 2 G x,y G Figure 1 The splitting operation has important applications in graph theory. Fleischner [13] used this operation to characterize Eulerian graphs and also gave an algorithm in terms of this operation to find all distinct Eulerian trails in an Eulerian graph. Tutte [40] characterized 3-connected graphs, and Slater [38] classified 4-connected graphs using a slight variation of this operation. Suppose G is a graph with n vertices and m edges. The relation between the incidence matrices of G and G x,y is noted as follows. Let {x, y, x 1, x 2,, x k } be a subset of E(G) incident at vertex v. The incident matrix A of G is a matrix of size n m. The row corresponding to the vertex v has entry 1 in the columns of x, y, x 1, x 2,, x k and 0 in the other columns. The graph G x,y has (n + 1) vertices and m edges. The incidence matrix A x,y of G x,y is a matrix of size (n+1) m. The row corresponding to v has the entry 1 in the columns

3 2.1 Introduction 21 of x 1, x 2,, x k and 0 in the other columns, where as the row corresponding to the vertex v x,y has the entry 1 in the columns of x, y and 0 in other columns. One can check that the matrix A x,y can be obtained from A by adjoining an extra row (corresponding to the vertex v x,y ) to A with entries zero every where except in the columns corresponding to x, y where it takes the value 1. And the row vector corresponding to the vertex v in A x,y is obtained by addition(mod 2) of row vectors corresponding to vertices v and v x,y. Considering the above fact, Raghunathan, Shikare and Waphare [26] extended the splitting operation from graphs to binary matroids. The corresponding operation is defined for a pair of elements of a binary matroid in the following way: Definition Let M be a binary matroid on a set E and A be a matrix over GF (2) that represents the matroid M. Consider elements x and y of M. Let A x,y be the matrix that is obtained by adjoining an extra row to A with this row being zero everywhere except in the columns corresponding to x and y where it takes the value 1. Let M x,y be the matroid represented by the matrix A x,y. We say that M x,y is obtained from M by splitting the pair of elements x and y. Moreover, the transition from M to M x,y is called the splitting operation. The two elements x and y of the matroid M x,y are now in series. Shikare, Azadi and Waphare [35] observed that one can split a bunch of edges incident at a vertex in a graph as shown in figure 2. Thus, a generalized splitting operation is defined as follows. Definition Let G be a graph and v be a vertex of degree at least

4 2.1 Introduction 22 n + 1 in G. Let T = {e 1, e 2,..., e n } be a set of edges incident at v and let e i = vx i for i = 1, 2,..., n. Then splitting away the edges in T from the vertex v results in a new graph G T obtained from G by deleting the edges e 1, e 2,..., e n and adding a new vertex v T adjacent to x 1, x 2,..., x n. The transition from G to G T is called a generalized splitting operation with respect to the set T. The graph G can be retrieved from G T by identifying the vertices v and v T. We illustrate this construction in figure 2. x 1 x 2 x n y 1 v y 2 y k x 1 x 2 x n y 1 v T v y 2 y k G Figure 2 G T Shikare, Azadi and Waphare [35] extended the generalized splitting operation to binary matroids in the following way. Definition Let M = M[A] be a binary matroid with ground set E and suppose X is a subset of E. Let A X be the matrix obtained from A by adjoining an extra row to A with this row being zero everywhere except in the columns corresponding to the elements of X where it takes the value 1. The splitting matroid M X is defined to be the vector matroid of the matrix A X. The transition from M to M X is called a generalized splitting operation.

5 2.2 Properties of Splitting Matroids Properties of Splitting Matroids Let M be a matroid and X E(M). We assume that M is loopless and coloopless. The set of circuits of M is denoted by C (M). We call a circuit of M as an OX-circuit if it contains an odd number of elements of the set X. Shikare, Azadi and Waphare [35] characterized the circuits of the splitting matroid M X as follows. Lemma Let M be a binary matroid on E and suppose X E. Then C(M X ) = C 0 C 1 where C 0 = {C C(M) C contains an even number of elements of X }; and C 1 = The set of minimal members of {C 1 C 2 C 1, C 2 C(M), C 1 C 2 = φ and each of C 1 and C 2 is an OX-circuit such that C 1 C 2 contains no member of C 0 }. In the following lemma, Shikare, Azadi and Waphare [35] characterized the rank function of the matroid M X in terms of the rank function of the matroid M. Lemma Let r and r be the rank functions of the matroids M and M X, respectively. Suppose A E(M). Then r (A) = r(a) + 1 if A contains an OX-circuit of M; and (2.2.1) = r(a) otherwise (2.2.2) The splitting operation, in general, may not preserve the connectedness of the binary matroid. In the next result Shikare [33] provided

6 2.3 A Characterization of n-connected Splitting Matroids 24 a sufficient condition for the splitting operation to yield a connected binary matroid from a 4-connected binary matroid. Theorem Let M be a 4-connected binary matroid with E(M) 9 and let x, y be distinct elements of M. Then M x,y is connected binary matroid. Borse and Dhotre [5] strengthened Shikare s result as stated in the following Theorem. Theorem Let M be a connected and vertically 3-connected binary matroid and x, y be distinct elements of M. Suppose that every cocircuit Q of M containing x, y is of size at least 4 and further, Q does not contain a 2-circuit of M. Then M x,y is connected binary matroid. The condition is necessary but not sufficient (see Example 2.3.4). The following result provides a necessary condition for a matroid to be n-connected (see [24]). Lemma If M is an n-connected matroid and E(M) 2(n 1), then all circuits and all cocircuits of M have at least n elements. In the next section, we provide a necessary and sufficient condition for an n-connected binary matroid M under which M X is n-connected. 2.3 A Characterization of n-connected Splitting Matroids The generalized splitting operation on a connected binary matroid, in general, may not yield a connected binary matroid. If M is a connected

7 2.3 A Characterization of n-connected Splitting Matroids 25 binary matroid and X < 2, then X will be a cocircuit of M X of size less than 2 (see [20]). Therefore, by Lemma 2.2.5, M X is not connected. In the following lemma, we provide a necessary and sufficient condition for a connected binary matroid M under which M X is connected. Lemma Let M be a connected and vertically 3-connected binary matroid of girth at least 3, X E(M) with X 2. Then M X is connected binary matroid if and only if for every x E(M) there is an OX-circuit of M not containing x. Proof. To show that the condition is sufficient, suppose that M X is not connected. Let (A, B) be a 1-separation of E(M X ). Then min { A, B } 1 and r (A) + r (B) r (M X ) 0. (2.3.1) By Lemma 2.2.2, we have r (A) = r(a) + 1 if A contains an OX-circuit of M; and (2.3.2) = r(a) otherwise. (2.3.3) r (B) = r(b) + 1 if B contains an OX-circuit of M; and (2.3.4) = r(b) otherwise. (2.3.5) Now one of the following two cases occurs. Case (I) A = 1. Suppose A = {z} where z E(M). Now M contains an OX-circuit C of M and {z} C = φ, implies that C B. Then, by Lemma 2.2.2, r (B) = r(b) + 1. And, by inequality (2.3.1), we have r(a) + r(b) + 1 r(m) 1 0.

8 2.3 A Characterization of n-connected Splitting Matroids 26 That is, r(a) + r(b) r(m) 0, and A, B 1. Hence (A, B) is a 1-separation of M, a contradiction. Case (II) A, B > 1. We apply Lemma to A and B. Suppose, each of A and B contains an OX-circuit of M. Then, by inequalities (2.3.1), (2.3.2) and (2.3.4), r(a) r(b) + 1 r(m) 1 0, that is, r(a) + r(b) r(m) 1 and A, B 1. We conclude that M has a one separation, a contradiction. Let exactly one of A and B contains an OX-circuit of M. Then, by inequalities (2.3.1), (2.3.3) and (2.3.4), r(a) + r(b) + 1 r(m) 1 0. And by inequalities (2.3.1), (2.3.2) and (2.3.5), r(a) r(b) r(m) 1 0 that is, r(a) + r(b) r(m) 0 and A, B 1. This leads to a 1-separation of M, a contradiction. Assume that neither A nor B contains an OX-circuit of M. Then, by inequalities (2.3.1), (2.3.3) and (2.3.5), r(a) + r(b) r(m) 1 0, Consequently, r(a) + r(b) r(m) 1 and since girth of M is at least 3, r(a), r(b) 2. Thus, we get a vertical 2-separation of M, a contradiction. This implies that M X has no 1-separation. We conclude that M X is a connected binary matroid.

9 2.3 A Characterization of n-connected Splitting Matroids 27 To check the necessity of the condition, suppose that M X is connected for X E(M) where X 2. On the way of contradiction, assume that there is an element x E(M) which is contained in every odd circuit of M. Let A = {x} and B = E(M)\A. Then B contains no OXcircuit of M. Thus, r (A) = 1, r (B) = r(b) = r(m) and A, B 1. Further, r (A) + r (B) r (M X ) = 1 + r(b) r(m) 1 = 0. Thus (A, B) forms a 1-separation of M X ; a contradiction. Consequently, we conclude that for every x E(M) there is an OX-circuit of M not containing x. This completes the proof. Remark Let M be an n-connected binary matroid. If X < n, then X will be a cocircuit of M X of size less than n (see [20]). So, by Lemma 2.2.5, M X is not connected. We generalize Lemma to n-connected binary matroids. In fact by the following theorem, we characterize n-connected binary matroids which yields n-connected binary matroids under the generalized splitting operation. Theorem Let M be an n-connected and vertically (n + 1)- connected binary matroid, n 2, E(M) 2(n 1) and girth of M is at least n + 1. Let X E(M) with X n. Then M X is n- connected if and only if for any (n 1)-element subset S of E(M) there is an OX-circuit C of M such that S C = φ. Proof. The proof is by induction on n. We proved the case n = 2 in Lemma Assume that the result is true for some integer k 2. Then we prove that the result is true for the integer k + 1.

10 2.3 A Characterization of n-connected Splitting Matroids 28 Let M be a (k+1)-connected and (k+2)-vertically connected binary matroid and M X be the splitting matroid arising from M. Suppose that M is of girth at least (k + 2) and for any k- element subset S of E(M), there is an OX-circuit C of M such that S C = φ. Then M X is k-connected by induction hypothesis. Thus, it is enough to show that M X has no k-separation. On the contrary, suppose M X is not (k + 1)-connected. Let (A, B) be a k-separation of E(M X ). Then min { A, B } k, and r (A) + r (B) r (M X ) k 1. (2.3.6) By Lemma 2.2.2, we have r (A) = r(a) + 1 if A contains an OX-circuit of M; and (2.3.7) = r(a) otherwise. (2.3.8) r (B) = r(b) + 1 if B contains an OX-circuit of M; and(2.3.9) = r(b) otherwise. (2.3.10) Now one of the following two cases occurs. Case (I) A = k. Let A = {x 1, x 2,...x k } where x i E(M) for i = 1 to k. As M is (k + 1)-connected, A does not contain any circuit of M. So, by Lemma 2.2.2, we have r (A) = r(a). Let S = {x 1, x 2,...x k }. Then there is an OX-circuit C of M such that S C = φ. This implies that C B and, by Lemma 2.2.2, r (B) = r(b) + 1. Hence, by inequalities (2.3.6), (2.3.8) and (2.3.9), we get r(a) + r(b) + 1 r(m) 1 k 1,

11 2.3 A Characterization of n-connected Splitting Matroids 29 that is, r(a) + r(b) r(m) k 1 with A, B k. This implies that (A, B) is a k-separation of M, a contradiction. Case (II) A, B > k. We apply Lemma to A and B. Let each of A and B contains an OX-circuit of M. Then, by inequalities (2.3.6), (2.3.7) and (2.3.9), r(a) r(b) + 1 r(m) 1 k 1, that is, r(a) + r(b) r(m) k 2 where A, B k. We conclude that M has a (k 1)-separation, a contradiction. Now suppose exactly one of A and B contains an OX-circuit of M. Then, by inequalities (2.3.6), (2.3.8) and (2.3.9), r(a) + r(b) + 1 r(m) 1 k 1. And by inequalities (2.3.6), (2.3.7) and (2.3.10), r(a) r(b) r(m) 1 k 1 that is, r(a) + r(b) r(m) k 1 and A, B k. This leads to a k-separation of M, a contradiction. Assume that neither A nor B contains an OX-circuit of M. Then, by inequalities (2.3.6), (2.3.8) and (2.3.10), r(a) + r(b) r(m) 1 k 1, Consequently, r(a) + r(b) r(m) k and since M is of girth at least (k + 2), r(a), r(b) k + 1. This indicates that (A,B) is a vertical (k + 1)-separation of M, a contradiction.

12 2.3 A Characterization of n-connected Splitting Matroids 30 Thus, M X has no k-separation. We conclude that M X is (k + 1)- connected. Consequently, by principle of mathematical induction, the result is true for all n 2. To check the necessity of the condition, suppose M X is (k + 1)- connected for X E(M) with X (k + 1). On the contrary, assume that there is a k- element subset S of E(M) such that for any OXcircuit C of M, S C φ. Let A = S and B = E(M) \ A. As M is (k + 1)-connected, we have r(a) + r(b) r(m) = k. By assumption B does not contain any OX-circuit of M. Therefore, r (A) = r(a), r (B) = r(b) and A, B k. Further, r (A) + r (B) r (M X ) = r(a) + r(b) r(m) 1 = k 1. Thus (A, B) forms a k-separation of M X, a contradiction. We conclude that for any k-element subset S of E(M) there is an OX-circuit C of M such that S C φ. This completes the proof. Example (Fano Matroid). We illustrate Lemma and Theorem with the help of Fano Matroid F 7 (see Example ). Consider the following representation of F 7 over GF (2) A = The set of all circuits of F 7 is C(F 7 ) = {{1, 2, 4}, {1, 3, 6}, {1, 7, 5}, {4, 3, 7}, {4, 6, 5}, {5, 3, 2}, {2, 6, 7}, {3, 5, 6, 7}, {2, 4, 5, 7}, {2, 3, 4, 6}, {1, 2, 5, 6}, {1, 2, 3, 7}, {1, 4, 6, 7},

13 2.3 A Characterization of n-connected Splitting Matroids 31 {1, 3, 4, 5}}. Let X = {1, 2}. Observe that for every z E(F 7 ) there is an OXcircuit of F 7 not containing z. Then the set of all circuits of splitting matroid (F 7 ) X is given by the set C((F 7 ) X ) = {{1, 2, 4}, {4, 3, 7}, {4, 6, 5}, {3, 5, 6, 7}, {1, 2, 5, 6}, {1, 2, 3, 7}}. It follows that, the splitting matroid (F 7 ) X is connected. Further, the matroid F 7 is 3-connected and vertically 4-connected binary matroid. If X = {1, 2, 4}, then for every pair {x, y} of elements of F 7 there is an OX-circuit of M not containing x and y. One can verify that (F 7 ) X is 3-connected. In the next example, we verify the necessity of condition in Theorem and Theorem Example Let M = M(K 4 ) be a cycle matroid of K 4 (see Figure 3). Let X = {x, y} be a pair of adjacent edges. Note that M is connected but there is no OX-circuit of M which avoids edge z. Thus, the condition of Lemma is not satisfied and M X is not connected. z x y Figure 3

14 2.4 A Minor based Characterization of n-connected Matroids A Minor based Characterization of n-connected Matroids In this section, we obtain a characterization of n-connected matroids. It is well known that a matroid is binary if and only if it has no minor isomorphic to U 2,4, uniform matroid of rank 2 on 4 element set. Extending this result to non binary matroids, Bixby proved that every element in a non-binary connected matroid is in a U 2,4 -minor. The result was further extended by Seymour [37] to non binary 3-connected matroids. Indeed, He proved the following theorem. Theorem Let M be a non binary 3-connected matroid and suppose that {a, b} E(M). Then M has a U 2,4 -minor using {a, b}. Suppose r 3. The wheel W r of order r is a graph having r + 1 vertices, r of which lie on a cycle (the rim); the remaining vertices are joined by a single edge (a spoke) to each of the other vertices. The whirl W r or order r is a matroid on E(W r ) having as its circuits all cycles of W r other than rim, as well as all sets of edges formed by adding a single spoke to the set of edges of the rim. In the following Theorem, Oxley [25], further, generalized Seymour s result within the class of non binary 3-connected matroids. Theorem Let M be a non binary 3-connected matroid and suppose that {a, b, c} E(M). Then either M has a U 2,4 -minor using {a, b, c}, or M has a W 3 -minor in which {a, b, c} is its rim or the set of spokes.

15 2.4 A Minor based Characterization of n-connected Matroids 33 The next result of Oxley [25] corresponds to Theorem in the case that one does not restrict to non-binary matroids. Theorem Let M be a 3-connected matroid having rank and corank at least three and suppose that {a, b, c} E(M). Then M has a minor isomorphic to one of U 3,6, P 6, Q 6, W 3, or M(K 4 ) that uses {a, b, c}. Tyler Moss [39] extended Theorem in the following result. Theorem A matroid M on at least four elements is 3-connected if and only if for each 4-element subset S of E(M), there is a minor N of M such that S E(N), and N is isomorphic to one of W 2, W 3, W 4, Q 6, M(W 3 ), M(W 4 ). We give a more general result for n-connected matroids as follows. Theorem Let M be a matroid and E(M) 2n 1. Then M is n-connected if and only if for every (2n 2)-element subset S of E(M) there is an n-connected minor N of M such that S E(N). Proof. Let M be an n-connected matroid. Then for every (2n 2)- element subset S of E(M), we can take N = M, so that N is n- connected minor of M and S E(N). Conversely, suppose that for every (2n 2)-element subset S of E(M), there is an n-connected minor N of M such that S E(N). Then we show that M is n-connected. On the contrary, assume that M is not n-connected. Then there is an (n 1)-separation (A, B) of M. That is, r(a) + r(b) r(m) n 2 (ii)

16 2.4 A Minor based Characterization of n-connected Matroids 34 where A, B n 1. Let S = {a 1, a 2, a 3,...a n 1 } {b 1, b 2, b 3,...b n 1 } where {a 1, a 2, a 3,...a n 1 } A and {b 1, b 2, b 3,...b n 1 } B. Then by hypothesis, there is an n-connected minor N of M such that S E(N). If X = A N and Y = B N, then X Y = N and X, Y n 1. Since N is n-connected, r(x) + r(y ) r(n) > n 2. (iii) Now, X A, Y B, so, by Lemma and inequality (iii), we have r(a) + r(b) r(m) r(x) + r(y ) r(n) > n 2. This is a contradiction in the light of inequality (ii). So, we conclude that M can not have (n 1) separation and hence it is n-connected. Remark When n = 2, we obtain Proposition as a corollary to Theorem * * * * * * *