Recognising Graphic and Matroidal Connectivity Functions

A {\em connectivity function} on a set $E$ is a function $λ:2^E\rightarrow \mathbb R$ such that $λ(\emptyset)=0$, that $λ(X)=λ(E-X)$ for all $X\subseteq E$, and that $λ(X\cap Y)+λ(X\cup Y)\leq λ(X)+λ(Y)$ for all $X,Y \subseteq E$. 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.

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