Mobility analysis is one of the fundamental problems of structural studies of kinematic chains. Degeneracy testing, an important step in structural synthesis, can be considered as a part of the mobility analysis due to the similarity of the two problems. One common error in the algorithms for these two problems is the assumption that the graph of a planar kinematic chain is a planar graph. This work shows that almost all the mobility analysis algorithms, except that of Lee and Yoon, have this error. This work also critically reviews the two most efficient algorithms on degeneracy testing, those by Hwang and Hwang, and Lee and Yoon. It is shown that due to the errors in the Hwang and Hwang’s algorithm, it failed to identify some of the degenerate chains. Furthermore, the accuracy of the Lee and Yoon’s algorithms for mobility analysis and degeneracy testing is proved by providing the mathematical justification of the individual steps of the algorithms.