The complexity class Unique End of Potential Line (UniqueEOPL) was introduced in 2018 by Fearnley et al. and captures total search problems that are promised to have a unique solution. Therefore, the original promise version of each problem is transformed to a total search version. UniqueEOPL contains some interesting problems like P-LCP, Cube-USO and Arrival. It is an especially interesting class because it has a  ”natural” complete problem: One Permutation Discrete Contraction (OPDC).

The goal of this work is to give a comprehensive introduction to the complexity theory of search problems with a focus on the class UniqueEOPL. A list of all currently known problems in UniqueEOPL is presented and for each problem, a definition and examples are given. Some selected reductions are shown in full detail and with corrections.

The main contribution is the new containment result of Grid-USO in UniqueEOPL, which is proven via a reduction from Grid-USO to Unique Forward EOPL. From that result it also follows that P-GLCP is contained in UniqueEOPL too.