  RNA design is the search for a sequence or set of sequences that will fold
into predefined structures, also known as the inverse problem of RNA folding.
While numerous RNA design methods have been invented to find sequences capable
of folding into a target structure, little attention has been given to the
identification of undesignable structures according to the minimum free energy
(MFE) criterion under the Turner model. In this paper, we address this gap by
first introducing mathematical theorems outlining sufficient conditions for
recognizing undesignable structures, then proposing efficient algorithms,
guided by these theorems, to verify the undesignability of RNA structures.
Through the application of these theorems and algorithms to the Eterna100
puzzles, we demonstrate the ability to efficiently establish that 15 of the
puzzles indeed fall within the category of undesignable structures. In
addition, we provide specific insights from the study of undesignability, in
the hope that it will enable more understanding of RNA folding and RNA design.
