Errors in graph embedding algorithms

One major area of difficulty in developing an algorithm for embedding a graph on a surface is handling bridges which have more than one possible placement. This paper addresses a number of published algorithms where this has not been handled correctly. This problem arises in certain presentations of the Demoucron, Malgrange and Pertuiset planarity testing algorithm. It also occurs in an algorithm of Filotti for embedding 3-regular graphs on the torus. The same error appears in an algorithm for embedding graphs of arbitrary genus by Filotti, Miller and Reif. It is also present in an algorithm for embedding graphs of arbitrary genus by Djidjev and Reif. The omission regarding the Demoucron, Malgrange and Pertuiset planarity testing algorithm is easily remedied. However there appears to be no way of correcting the algorithms of the other papers without making the algorithms take exponential time.