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Generalization of Pinching Operation to Binary Matroids J. Algebra Comb. Discrete Appl. Received: 14 April 2020 7(3) • 247–258 Accepted: 1 May 2020 Journal of Algebra Combinatorics Discrete Structures and Applications Generalization of pinching operation to binary matroids Research Article Vahid Ghorbani, Ghodratollah Azadi, Habib Azanchiler Abstract: In this paper, we generalize the pinching operation on two edges of graphs to binary matroids and investigate some of its basic properties. For n ≥ 2, the matroid that is obtained from an n-connected matroid by this operation is a k-connected matroid with k 2 f2; 3; 4g or is a disconnected matroid. We find conditions to guarantee this k. Moreover, we show that Eulerian binary matroids are char- acterized by this operation and we also provide some interesting applications of this operation. 2010 MSC: 05B35 Keywords: Binary matroid, Connectivity, Pinching, Splitting, Splitting off, Element splitting 1. Introduction The matroid and graph notations and terminology used here will follow [4] and [7]. Let G be a graph p and S be a subset of the edge set E(G). We denote by GS the graph that is obtained from G by adding p a new vertex w and replacing any edge uv 2 S with two edges uw and wv. The transition from G to GS is called the k-pinching operation [2] where k = jSj. If S is the empty set, then 0-pinching means adding a new single vertex w. If jSj = 1, then applying the pinching operation is equivalent to the subdivision p of uv where S = fuvg. If S = E(G), then we have the parallel classes on the new vertex w on GS such that the sequence of the cardinality of them is equal to the degree sequence of the vertices of G. Applying the pinching operation on graphs is a useful method for solving 2k-edge-connectivity prob- lems for graphs [2]. For example, suppose that G is a 2k-edge-connected graph. Then the graph G0 obtained from G by pinching together any k edges of G is also 2k-edge-connected. Raghunathan, Shikare, and Wapare [5] extended the splitting operation and Azadi [1] extended the splitting off and the element splitting operations from graphs to binary matroids. These operations are defined as follows. Definition 1.1. Let M be a binary matroid on a set E and A be a matrix that represents M over GF (2). Consider two elements x and y of E(M). Let Ax;y be the matrix that is obtained by adjoining an extra Vahid Ghorbani (Corresponding Author), Ghodratollah Azadi, Habib Azanchiler; Department of Mathematics, Urmia University, Iran (email: [email protected], [email protected], [email protected]) http://dx.doi.org/10.13069/jacodesmath.784992 ISSN 2148-838X 247 V. Ghorbani et al. / J. Algebra Comb. Discrete Appl. 7(3) (2019) 247–258 row to A whose entries are zero everywhere except in the columns corresponding to x and y where it takes value 1. Let Mx;y be the matroid that represents by the matrix Ax;y. Then the transition from M to Mx;y is called the splitting operation. Definition 1.2. Let M be a binary matroid on a set E and A be a matrix that represents M over GF (2). 0 Consider two elements x and y of E(M). Let Ax;y be the matrix that is obtained by adjoining an extra row to A whose entries are zero everywhere except in the columns corresponding to x and y where it takes value 1 and then adjoining an extra column to the resulting matrix with this column being zero everywhere 0 0 except in the last row. Let Mx;y be the matroid that represents by the matrix Ax;y. Then the transition 0 from M to Mx;y is called the element splitting operation. Definition 1.3. Let M be a binary matroid on a set E and A be a matrix that represents M over GF (2). Consider two elements x and y of E(M). Let Axy be the matrix that is obtained by adjoining an extra column to A which is the sum of the columns corresponding to x and y, and then deleting the two columns corresponding to x and y. Let Mxy be the matroid that represents by the matrix Axy. Then the transition from M to Mxy is called the splitting off operation. Splitting off operation on graphs is defined as follows. Definition 1.4. Let G be a graph. Consider two adjacent non-loop edges x = uw and y = vw. The splitting off a pair (x; y) of edges from a vertex w means that we replace the edges x and y by a new edge α = uv. If u = v then the resulting loop is deleted from the graph. 2. 2-pinching operation in 2-connected graphs Let G be a 2-connected graph with the edge set E(G) and S ⊆ E(G) where S = fx = u1v1; y = u2v2g. To apply the 2-pinching operation, we first delete x and y and add new vertex w. Then we add new 0 00 0 00 edges x = u1w, x = wv1, y = u2w and y = wv2. When x and y are adjacent, we take v1 = v2 = v. 0 00 0 00 p Therefore, x = u1w, x = wv, y = u2w and y = wv (Figure1). We denote by Gxy the graph that is obtained by applying the 2-pinching operation on edges x and y. Note that we can retrieve the graph G p 0 00 0 00 from Gxy by splitting off the pair (x ; x ) and then splitting off the pair (y ; y ). Figure 1. 2-pinching operation on fx; zg and fx; yg. Now let G be a 2-connected graph and x; y 2 E(G). Then G has a cycle Cn of length n containing both x and y. Assume that the cycle Cn has the minimum cardinality among all such cycles. After 0 0 applying the 2-pinching operation on x and y, the cycle Cn transforms into two cycles Ck+2 and Cn−k 00 00 0 0 0 where k is the length of shortest path Q among endpoints of x and y such that x ; y 2 Ck+2 and x ; y 2 0 0 00 00 1 0 0 0 Cn−k. In other words, Ck+2 = E(Q)[fx ; y g and Cn−k = Cn −(E(Q)[fx; yg) [fx ; y g. Note that 0 00 00 0 0 0 if k = 0, then x and y are adjacent and Cn transforms to C2 = fx ; y g and Cn = Cn −fx; yg [fx ; y g. 1 In this section, for cycle Cn of length n from a graph G, we consider just its edge set 248 V. Ghorbani et al. / J. Algebra Comb. Discrete Appl. 7(3) (2019) 247–258 Theorem 2.1. Let G be a graph and x; y 2 E(G). Suppose that G has a cycle Cn with a minimum 0 00 0 00 0 0 length containing both x and y. Let x ; x ; y ; y ,Ck+2 and Cn−k be the notations as mentioned above. 0 0 p p Moreover, let δ = fCk+2;Cn−kg and C be a subset of edges of Gxy. Then C forms a cycle of Gxy if and only if C 2 δ or C satisfies one of the following conditions. (i) C is a cycle of G which contains neither x nor y; (ii) C = C0−fxg[fx0; x00g where C0 is a cycle of G containing x but not y, or C = C00−fyg[fy0; y00g where C00 is a cycle of G containing y but not x; and (iii) C = C1∆C2 where C1 2 δ, and C2 is a cycle of type (i) or (ii) such that C1 \ C2 6= ;. Proof. It is clear that if C 2 δ or if C is a cycle of G containing neither x nor y, then C is a cycle of p 0 0 00 0 0 Gxy. Now suppose that C = C − fxg [ fx ; x g = C − fu1v1g [ fu1w; v1wg where C is a cycle of G containing x but not y and has length k0 + 1, so 0 0 C = v1xu1e1z1:::ek0−1zk0−1ek0 v1: 0 00 After applying the pinching operation on x we delete x and add x = u1w and x = v1w. Thus 0 0 00 00 0 (C − fxg) [ fx ; x g = u1e1z1:::ek0−1zk0−1ek0 v1x wx u1: p 00 0 00 0 Therefore C is a cycle of Gxy. Similarly, if C = C −fyg [fy ; y g = C −fu2v2g [fu2w; v2wg where 00 p C is a cycle of G containing y but not x, then C is a cycle of Gxy. Now suppose that C1 is a cycle of p type (i) or (ii) and C2 2 δ such that C1 \ C2 6= ;. To show that C1∆C2 is a cycle of Gxy, we consider the following two possible cases. Figure 2. Internally disjoint paths. 0 Case(1) Let C1 be a cycle of type (i). Assume that C1 has a non-empty intersection with both Ck+2 and 0 0 0 Cn−k. Let Q1 and Q2 be the paths such that C1\Cn−k = E(Q1) and C1\Ck+2 = E(Q2). Moreover, let Q3 , Q4 be internally disjoint paths which have the same end vertices as of Q1 and Q2 such that 0 C1 = E(Q1)[E(Q2)[E(Q3)[E(Q4) (See figure2).
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