Recognising graphic and matroidal connectivity functions

A connectivity function on a set E is a function lambda : 2(E) -> R such that lambda(0) = 0, that lambda(X) = lambda(E-X) for all X subset of E, and that lambda(X boolean AND Y)+lambda(X boolean OR Y) <= lambda(X)+ lambda(Y) for all X, Y subset of E. Graphs, matroids and, more generally, polymatroids have associated connectivity functions. In this paper we give a method for identifying when a connectivity function comes from a graph. This method uses no more than a polynomial number of evaluations of the connectivity function. In contrast, we show that the problem of identifying when a connectivity function comes from a matroid cannot be solved in polynomial time. We also show that the problem of identifying when a connectivity function is not that of a matroid cannot be solved in polynomial time. (c) 2020 Elsevier B.V. All rights reserved.