Essential elements in connected k-polymatroids

It is a well-known result of Tutte that, for every element x of a connected matroid M, at least one of the deletion and contraction of x from M is connected. This paper shows that, in a connected k-polymatroid, only two such elements are guaranteed. We show that this bound is sharp and characterize those 2-polymatroids that achieve this minimum. To this end, we define and make use of a generalized parallel connection for k-polymatroids that allows connecting across elements of different ranks. This study of essential elements gives results crucial to finding the unavoidable minors of connected 2-polymatroids, which will appear elsewhere.