Simple DFS on the Complement of a Graph and on Partially Complemented Digraphs

A complementation operation on a vertex of a digraph changes all outgoing arcs into non-arcs, and outgoing non-arcs into arcs. A partially complemented digraph G̃ is a digraph obtained from a sequence of vertex complement operations on G. Dahlhaus et al. showed that, given an adjacency-list representation of G̃, depth-first search (DFS) on G can be performed in O(n+ m̃) time, where n is the number of vertices and m̃ is the number of edges in G̃. To achieve this bound, their algorithm makes use of a somewhat complicated stack-like data structure to simulate the recursion stack, instead of implementing it directly as a recursive algorithm. We give a recursive O(n + m̃) algorithm that uses no complicated data-structures.