Some Results Involving the Splitting Operation on Binary Matroids

We obtain some results concerning the planarity and graphicness of the splitting matroids. Further, we explore the effect of splitting operation on the sum of two matroids. 1. Introduction The matroid notations and terminology used here will follow Oxley [1]. Fleischner [2] defined the splitting operation for a graph with respect to a pair of adjacent edges as follows. Let be a connected graph and let be a vertex of degree at least three in . If and are two edges incident at , then splitting away the pair from results in a new graph obtained from by deleting the edges and , and adding a new vertex adjacent to and . The transition from to is called the splitting operation on . Figure 1 illustrates this construction explicitly. Figure 1 Fleischner [2] used the splitting operation to characterize Eulerian graphs. Fleischner [3] also developed an algorithm to find all distinct Eulerian trails in an Eulerian graph using the splitting operation. Tutte [4] characterized 3-connected graphs, and Slater [5] classified 4-connected graphs using a slight variation of this operation. Raghunathan et al. [6] extended the splitting operation from graphs to binary matroids as follows. Definition 1.1. Let be a binary matroid and suppose . Let be the matrix obtained from by adjoining the row that is zero everywhere except for the entries of in the columns labeled by and . The splitting matroid is defined to be the vector matroid of the matrix . Alternatively, the splitting operation can be defined in terms of circuits of binary matroids as follows. Lemma 1.2 (see [6]). Let be a binary matroid on a set together with the set of circuits and let . Then with , where or ; and and contains no member of . Shikare and Waphare [7] characterized graphic matroids whose splitting matroids are also graphic. In fact, they proved the following theorem. Theorem 1.3 (see [7]). The splitting operation, by any pair of elements, on a graphic matroid yields a graphic matroid if and only if the matroid has no minor isomorphic to the cycle matroid of any of the four graphs of Figure 2. Figure 2: Forbidden graphs. We define a matroid to be planar if it is both graphic and cographic. Let us consider the graph (i) of Figure 2. In this graph by splitting away two non-adjacent elements and , we will get a matroid which is not planar. This example exhibits the fact that even if the original matroid is planar, there exist pairs of non-adjacent elements and such that the splitting matroid is not planar. Now, by considering above example, we prove the following Theorem. Theorem 1.4. Let be a planar