  For a class $\mathcal C$ of graphs $G$ equipped with functions $f_G$ defined
on subsets of $E(G)$ or $V(G)$, we say that $\mathcal{C}$ is $k$-scattered with
respect to $f_G$ if there exists a constant $\ell$ such that for every graph
$G\in \mathcal C$, the domain of $f_G$ can be partitioned into subsets of size
at most $k$ so that the union of every collection of the subsets has $f_G$
value at most $\ell$. We present structural characterizations of graph classes
that are $k$-scattered with respect to several graph connectivity functions.
  In particular, our theorem for cut-rank functions provides a rough structural
characterization of graphs having no $mK_{1,n}$ vertex-minor, which allows us
to prove that such graphs have bounded linear rank-width.
