Partially Complemented Representations of Digraphs

A complementation operation on a vertex of a digraph changes all outgoing arcs into non-arcs, and outgoing non-arcs into arcs. This defines an equivalence relation where two digraphs are equivalent if one can be obtained from the other by a sequence of such operations. We show that given an adjacency-list representation of a digraph G, many fundamental graph algorithms can be carried out on any member G′ of G’s equivalence class in O(n+m) time, where m is the number of arcs in G, not the number of arcs in G′. This may have advantages when G′ is much larger than G. We use this to generalize to digraphs a simple O(n + m logn) algorithm of McConnell and Spinrad for finding the modular decomposition of undirected graphs. A key step is finding the strongly-connected components of a digraph F in G’s equivalence class, where F may have ω(m logn) arcs.