# 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).

## Definition

The cycle rank r(G) of a digraph G = (VE) is inductively defined as follows:

$r(G) = 1 + \min_{v\in V} r(G-v),\,$ where G - v is the digraph resulting from deletion of vertex v and all edges beginning or ending at v.
• If G is not strongly connected, then r(G) is equal to the maximum cycle rank among all strongly connected components of G.

## History

Cycle rank was introduced by Eggan (1963) in the context of star height of regular languages. It was rediscovered by (Eisenstat & Liu 2005) as a generalization of undirected tree-depth, which had been developed beginning in the 1980s and applied to sparse matrix computations (Schreiber 1982).

## Examples

The cycle rank of a directed acyclic graph is 0, while a complete digraph of order n with a self-loop at each vertex has cycle rank n. Apart from these, the cycle rank of a few other digraphs is known: the undirected path $P_n$ of order n, which possesses a symmetric edge relation and no self-loops, has cycle rank $\lfloor \log n\rfloor$ (McNaughton 1969). For the directed $(m\times n)$-torus $T_{m,n}$, i.e., the cartesian product of two directed circuits of lengths m and n, we have $r(T_{n,n}) = n$ and $r(T_{m,n}) = \min\{m,n\} + 1$ for m ≠ n (Eggan 1963, Gruber & Holzer 2008).

## Computing the cycle rank

Computing the cycle rank is computationally hard: Gruber (2012) proves that the corresponding decision problem is NP-complete, even for sparse digraphs of maximum outdegree at most 2. On the positive side, the problem is solvable in time $O(1.9129^n)$ on digraphs of maximum outdegree at most 2, and in time $O^*(2^n)$ on general digraphs. There is an approximation algorithm with approximation ratio $O((\log n)^\frac32)$.

## Applications

### Star height of regular languages

The very first application of cycle rank was in formal language theory, for studying the star height of regular languages. Eggan (1963) established a relation between the theories of regular expressions, finite automata, and of directed graphs. In subsequent years, this relation became known as Eggan's theorem, cf. Sakarovitch (2009). In automata theory, a nondeterministic finite automaton with ε-moves (ε-NFA) is defined as a 5-tuple, (Q, Σ, δ, q0, F), consisting of

• a finite set of states Q
• a finite set of input symbols Σ
• a set of labeled edges δ, referred to as transition relation: Q × (Σ ∪{ε}) × Q. Here ε denotes the empty word.
• an initial state q0Q
• a set of states F distinguished as accepting states FQ.

A word w ∈ Σ* is accepted by the ε-NFA if there exists a directed path from the initial state q0 to some final state in F using edges from δ, such that the concatenation of all labels visited along the path yields the word w. The set of all words over Σ* accepted by the automaton is the language accepted by the automaton A.

When speaking of digraph properties of a nondeterministic finite automaton A with state set Q, we naturally address the digraph with vertex set Q induced by its transition relation. Now the theorem is stated as follows.

Eggan's Theorem: The star height of a regular language L equals the minimum cycle rank among all nondeterministic finite automata with ε-moves accepting L.

Proofs of this theorem are given by Eggan (1963), and more recently by Sakarovitch (2009).

### Cholesky factorization in sparse matrix computations

Another application of this concept lies in sparse matrix computations, namely for using nested dissection to compute the Cholesky factorization of a (symmetric) matrix in parallel. A given sparse $(n\times n)$-matrix M may be interpreted as the adjacency matrix of some symmetric digraph G on n vertices, in a way such that the non-zero entries of the matrix are in one-to-one correspondence with the edges of G. If the cycle rank of the digraph G is at most k, then the Cholesky factorization of M can be computed in at most k steps on a parallel computer with $n$ processors (Dereniowski & Kubale 2004).