Transitive Reduction

In mathematics, a transitive reduction of a directed graph is a graph with as few edges as possible that has the same reachability relation as the given graph. Equivalently, the given graph and its transitive reduction should have the same transitive closure as each other, and its transitive reduction should have as few edges as possible among all graphs with this property. Transitive reductions were introduced by Aho, Garey & Ullman (1972), who provided tight bounds on the computational complexity of constructing them.

If a given graph is a finite directed acyclic graph, its transitive reduction is unique, and is a subgraph of the given graph. However, uniqueness is not guaranteed for graphs with cycles, and for infinite graphs not even existence is guaranteed. The closely related concept of a minimum equivalent graph is a subgraph of the given graph that has the same reachability relation and as few edges as possible. For finite directed acyclic graphs, the minimum equivalent graph is the same as the transitive reduction. However, for graphs that may contain cycles, minimum equivalent graphs are NP-hard to construct, while transitive reductions can still be constructed in polynomial time. Transitive reductions can also be defined for more abstract binary relations on sets, by interpreting the pairs of the relation as arcs in a graph.