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Abstract: A standard matrix representation A of a matroid M represents M relative to a fixed basis B. Deleting rows and columns of A correspond to contracting elements of B and deleting elements of . If M is 3-connected, it is often desirable to perform such an element removal from M while maintaining 3-connectivity. This paper proves that this is always possible provided M has no 4-element fans. We also show that, subject to a mild essential restriction, this element removal can be done so as to retain a copy of a specified 3-connected minor of M. [Copyright &y& Elsevier]

Abstract: In this paper we prove that two quantities relating to the length of permutations defined on trees are independent of the structures of trees. We also find that these results are closely related to the results obtained by Graham and Pollak [R.L. Graham, H.O. Pollak, On the addressing problem for loop switching, Bell System Tech. J. 50 (1971) 2495–2519] and by Bapat, Kirkland, and Neumann [R. Bapat, S.J. Kirkland, M. Neumann, On distance matrices and Laplacians, Linear Algebra Appl. 401 (2005) 193–209]. [Copyright &y& Elsevier]