Extended Argumentation Frameworks (EAFs) are a recently proposed formalism that develop abstract argumentation frameworks (AFs) by allowing attacks between arguments to be attacked themselves: hence EAFs add a relationship [Dscr ] ⊆ [Xscr ] × [Ascr ] to the arguments ([Xscr ]) and attacks ([Ascr ] ⊆ [Xscr ] × [Xscr ]) in an AF's basic directed graph structure 〈[Xscr ],[Ascr ]〉. This development provides a natural way to represent and reason about preferences between arguments. Studies of EAFs have thus far focussed on acceptability semantics, proof-theoretic processes, and applications. However, no detailed treatment of their practicality in computational settings has been undertaken. In this paper we address this lacuna, considering algorithmic and complexity properties specific to EAFs. We show that (as for standard AFs) the problem of determining if an argument is acceptable w.r.t. a subset of [Xscr ] is polynomial time decidable and, thus, determining the grounded extension and verifying admissibility are efficiently solvable. We, further, consider the status of a number of decision questions specific to the EAF formalism in the sense that these have no counterparts within AFs.