The two basic linear time Planarity algorithms: Are they the same?

We compare the two most widely known planarity testing algorithms, the one due to Hoperoft and Tarjan, and the other to Lempel. Even, and Cederbaum. These two algorithms are generally viewed as very different approaches to the problems of planarity testing and graph embedding. In this paper, however, by utilizing previously unnoticed freedoms of choice in the order in which the operations of the Hoperoft-Tarjan algorithm can be performed, we create a variation of this algorithm which, in terms of the order in which vertices are processed, is indistinguishable on all planar graph inputs from the Lempel -Even-Cederbaum algorithm. This allows one to create hybridized algorithms which, when interpreted as embedding algorithms, combine all of the features of both procedures.