  We consider the Combinatorial RNA Design problem, a minimal instance of RNA
design where one must produce an RNA sequence that adopts a given secondary
structure as its minimal free-energy structure. We consider two free-energy
models where the contributions of base pairs are additive and independent: the
purely combinatorial Watson-Crick model, which only allows equally-contributing
A -- U and C -- G base pairs, and the real-valued Nussinov-Jacobson model,
which associates arbitrary energies to A -- U, C -- G and G -- U base pairs. We
first provide a complete characterization of designable structures using
restricted alphabets and, in the four-letter alphabet, provide a complete
characterization for designable structures without unpaired bases. When
unpaired bases are allowed, we characterize extensive classes of
(non-)designable structures, and prove the closure of the set of designable
structures under the stutter operation. Membership of a given structure to any
of the classes can be tested in $\Theta$(n) time, including the generation of a
solution sequence for positive instances. Finally, we consider a
structure-approximating relaxation of the design, and provide a $\Theta$(n)
algorithm which, given a structure S that avoids two trivially non-designable
motifs, transforms S into a designable structure constructively by adding at
most one base-pair to each of its stems.
