The Transitive Reduction of a Directed Graph

We consider economical representations for the path information in a directed graph. A directed graph G is said to be a transitive reduction of the directed graph G provided that (i) G has a directed path from vertex u to vertex v if and only if G has a directed path from vertex u to vertex v, and (ii) there is no graph with fewer arcs than G satisfying condition (i). Though directed graphs with cycles may have more than one such representation, we select a natural canonical representative as the transitive reduction for such graphs. It is shown that the time complexity of the best algorithm for finding the transitive reduction of a graph is the same as the time to compute the transitive closure of a graph or to perform Boolean matrix multiplication. 1. Introduction. Given a directed graph G, one is often interested in knowing whether there is a path from one vertex to another in that graph. In many cases it is possible to represent this information by another directed graph that has fewer arcs than the given graph. Informally, we say that a graph G is a transitive reduction of the directed graph G whenever the following two conditions are