In molecular biology, RNA structure comparison and motif search are of great interest for solving major problems such as phylogeny reconstruction, prediction of molecule folding and identification of common functions. RNA structures can be represented by arc-annotated sequences (primary sequence along with arc annotations), and this paper mainly focuses on the so-called arc-preserving subsequence (APS) problem where, given two arc-annotated sequences (S, P) and (T,Q), we are asking whether (T,Q) can be obtained from (S, P) by deleting some of its bases (together with their incident arcs, if any). In previous studies, this problem has been naturally divided into subproblems reflecting the intrinsic complexity of the arc structures. We show that APS(Crossing, Plain) is NP-complete, thereby answering an open problem posed in [11]. Furthermore, to get more insight into where the actual border between the polynomial and the NP-complete cases lies, we refine the classical subproblems of the APS problem in much the same way as in [19] and prove that both APS({⊏, ≬}, ∅) and APS({<, ≬}, ∅) are NP- complete. We end this paper by giving some new positive results, namely showing that APS({≬}, ∅) and APS({≬},{≬}) are polynomial time solvable.