Feedback vertex sets and cyclically reducible graphs

Abstract

The problem of finding a minimum cardinality feedback vertex set of a directed graph is considered. Of the classic NP-complete problems, this is one of the least understood. Although Karp showed the general problem to be NP-complete, a linear algorithm for its solution on reducible flow graphs was given by Shamir. The class of reducible flow graphs is the only nontrivial class of graphs for which a polynomial-time algorithm to solve this problem is known. The main result of this paper is to present a new class of graphs—the cyclically reducible graphs —for which minimum feedback vertex sets can be found in polynomial time. This class is not restricted to flow graphs, and most small graphs (10 or fewer nodes) fall into this class. The identification of this class is particularly important since there do not exist approximation algorithms for this problem having a provably good worst case performance. Along with the class and a simple polynomial-time algorithm for finding minimum feedback vertex sets of graphs in the class, several related results are presented. It is shown that there is no “forbidden subgraph” characterization of the class and that there is no particular inclusion relationship between this class and the reducible flow graphs. In addition, it is shown that a class of (general) graphs, which are related to the reducible flow graphs, are contained in the cyclically reducible class.