Topological sorting is, given with a directed acyclic graph G=(V,E), to find a total ordering of the vertices such that if (u,v)E then u is ordered before v. Instead of topological sorting, we are interested in how many total orderings exist in a given directed acyclic graph. We call such a total ordering as legal sequence and the problem of finding total number of legal sequences as legal sequence number problem. In this paper, we firstly give necessary definitions and known results obtained in our previous research. Then we give a method how to obtain legal sequence number for a class of directed acyclic graphs, extended 2-b-SPGs. Finally we discuss the complexity of legal sequence number problem for extended 2-b-SPGs.

Topological sorting is, given with a directed acyclic graph G = (V, E), to find a total ordering of the vertices such that if (u, v) E then u is ordered before v. Instead of finding total orderings, we wish to find out how many total orderings exist in a given directed acyclic graph G = (V, E). Here we call a total ordering as legal sequence and the problem as legal sequence number problem. In this paper, we first propose theorems on equivalent transformation of graphs with respect to legal sequence number. Then we give a formula to calculate legal sequence number of basic series-parallel digraphs and a way of the calculation for general series-parallel digraphs. Finally we apply our results to show how to obtain legal sequence number for a class of extended series-parallel digraphs.