The classes of Raspail (also known as Bipolarizable) and P4-simplicial graphs were introduced by Hoàng and Reed who showed that both classes are perfectly orderable and admit polynomial-time recognition algorithms HR1. In this paper, we consider the recognition problem on these classes of graphs and present algorithms that solve it in O(n m) time. In particular, we prove properties and show that we can produce bipolarizable and P4-simplicial orderings on the vertices of the input graph G, if such orderings exist, working only on P3s that participate in a P4 of G. The proposed recognition algorithms are simple, use simple data structures and both require O(n + m) space. Additionally, we show how our recognition algorithms can be augmented to provide certificates, whenever they decide that G is not bipolarizable or P4-simplicial; the augmentation takes O(n + m) time and space. Finally, we include a diagram on class inclusions and the currently best recognition time complexities for a number of perfectly orderable classes of graphs.