Deciding if a hyperbolic group splits over a given quasiconvex subgroup

We present an algorithm which decides whether a given quasiconvex residually finite subgroup H of a hyperbolic group G is associated with a splitting. The methods developed also provide algorithms for computing the number of filtered ends \tilde{e}(G,H) of H in G under certain hypotheses and give a new straightforward algorithm for computing the number of ends e(G,H) of the Schreier graph of H . Our techniques extend those of Barrett via the use of labelled digraphs, the languages of which encode information on the connectivity of \partial G - \Lambda H .