  We define and study a new structural parameter for directed graphs, which we
call \emph{ear anonymity}. Our parameter aims to generalize the useful
properties of \emph{funnels} to larger digraph classes. In particular, funnels
are exactly the acyclic digraphs with ear anonymity one. We prove that
computing the ear anonymity of a digraph is \NP/-hard and that it can be solved
in $O(m(n + m))$-time on acyclic digraphs (where \(n\) is the number of
vertices and \(m\) is the number of arcs in the input digraph). It remains open
where exactly in the polynomial hierarchy the problem of computing ear
anonymity lies, however for a related problem we manage to show
$\Sigma_2^p$-completeness.
