On the calculation of transitive reduction - closure of orders

Habib, H., M. Morvan and J.-X. Rampon, On the calculation of transitive reduction-closure of orders, Discrete Mathematics 111 (1993) 289-303. Computations of transitive closure and reduction ofdirected acyclic graphs are mainly considered in this paper. Classes of directed acyclic graphs for which such problems can be solved in linear time complexity (in accordance with the number of arcs) are proposed, namely: generalized N-free graphs, graphs such that the external or internal degree of any vertex is bounded in the transitive reduction, and bounded decomposition width orders, strongly W-free orders. For this purpose, we study the worst-case complexity of a nice algorithm. This algorithm is due to Goralcikova and Koubek (1979) and already studied by Mehlhorn (1984) and Simon (1988). Furthermore, two new classes of orders (generalized N-free and strongly W-free) which seem to have nice computational properties, are proposed and studied.