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We consider the Combinatorial RNA Design problem, a minimal instance of RNAdesign where one must produce an RNA sequence that adopts a given secondarystructure as its minimal free-energy structure. We consider two free-energymodels where the contributions of base pairs are additive and independent: thepurely combinatorial Watson-Crick model, which only allows equally-contributingA -- 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. Wefirst provide a complete characterization of designable structures usingrestricted alphabets and, in the four-letter alphabet, provide a completecharacterization for designable structures without unpaired bases. Whenunpaired bases are allowed, we characterize extensive classes of(non-)designable structures, and prove the closure of the set of designablestructures under the stutter operation. Membership of a given structure to anyof the classes can be tested in $\Theta$(n) time, including the generation of asolution sequence for positive instances. Finally, we consider astructure-approximating relaxation of the design, and provide a $\Theta$(n)algorithm which, given a structure S that avoids two trivially non-designablemotifs, transforms S into a designable structure constructively by adding atmost one base-pair to each of its stems.

Combinatorial RNA Design Designability and Structure-Approximating Algorithm in Watson-Crick and Nussinov-Jacobson Energy Models