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Spikes are an important class of 3-connected matroids. For an integer r ! 3, there is a unique binary r-spike denoted by Z r. When a circuit-hyperplane of Z r is relaxed, we obtain another spike and repeating this procedure will produce other non-binary spikes. The es-splitting operation on a binary spike of rank r, may not yield a spike. In this paper, we give a necessary and sufficient condition for the es-splitting operation to construct Z rþ1 directly from Z r. Indeed, all binary spikes and many of non-binary spikes of each rank can be derived from the spike Z 3 by a sequence of the es-splitting operations and circuit-hyperplane relaxations.

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Given a binary matroid M and a subset T ⊆ E(M ), Luis A. Goddyn posed a problem that the dual of the splitting of M , i.e., ((MT ) * ) is not always equal to the splitting of the dual of M , ((M * )T ). This persuade us to ask if we can characterize those binary matroids for which (MT ) * = (M * )T . Santosh B. Dhotre answered this question for a two-element subset T . In this paper, we generalize his result for any subset T ⊆ E(M ) and exhibit a criterion for a binary matroid M and subsets T for which (MT ) * and (M * )T are the equal. We also show that there is no subset T ⊆ E(M ) for which, the dual of element splitting of M , i.e., ((M ′ T ) * ) equals to the element splitting of the dual of M , ((M * ) ′ T ).

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, 3, 4} or is a disconnected matroid. We find conditions to guarantee this k. Moreover, we show that Eulerian binary matroids are characterized by this operation and we also provide some interesting applications of this operation.