Finding tracks in 2-complexes

We present an algorithm which given a presentation of a group G without 2-torsion, a solution to the word problem with respect to this presentation, and an acylindricity constant {\kappa}, outputs a collection of tracks in an appropriate presentation complex. We give two applications: the first is an algorithm which decides if G admits an essential free decomposition, the second is an algorithm which; if G is relatively hyperbolic; decides if it admits an essential elementary splitting.