Directed Ear Anonymity

We define and study a new structural parameter for directed graphs, which we call ear anonymity. Our parameter aims to generalize the useful properties of 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 \(\mathcal {O}(m(n + m))\) -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 \(\varSigma _2^p\) -completeness.