On reachability in graphs with bounded independence number

Abstract

We study the reachability problem for finite directed graphs whose independence number is bounded by some constant k. This problem is a generalisation of the reachability problem for tournaments. We show that the problem is first-order definable for all k. In contrast, the reachability problems for many other types of finite graphs, including dags and trees, are not first-order definable. Also in contrast, first-order definability does not carry over to the infinite version of the problem. We prove that the number of strongly connected components in a graph with bounded independence number can be computed using TC0-circuits, but cannot be computed using AC0-circuits. We also study the succinct version of the problem and show that it is ΠP 2 -complete for all k.