Cycle rank

In graph theory, the cycle rank of a directed graph is a digraph connectivity measure proposed first by Eggan and Büchi (Eggan 1963). Intuitively, this concept measures how close a digraph is to a directed acyclic graph (DAG), in the sense that a DAG has cycle rank zero, while a complete digraph of order n with a self-loop at each vertex has cycle rank n. The cycle rank of a directed graph is closely related to the tree-depth of an undirected graph and to the star height of a regular language. It has also found use in sparse matrix computations (see Bodlaender et al. 1995) and logic (Rossman 2008).