Finding a minimal transitive reduction in a strongly connected digraph within linear time

Abstract

This paper describes an algorithm for finding a minimal transitive reduction Gred of a given directed graph G, where Gred means a subgraph of G with the same transitive closure as G but itself not contains a proper subgraph G1 with the same property too. The algorithm uses depth-first search and two graph transformations preserving the transitive closure to achieve a time bound of O(n + m), where n stands for the number of vertices and m is the number of the edges.