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 Π P2 -complete for all k.
Work done in part while visiting the University of Rochester, New York. Supported by a TU Berlin Erwin-Stephan-Prize grant.
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Nickelsen, A., Tantau, T. (2002). On Reachability in Graphs with Bounded Independence Number. In: Ibarra, O.H., Zhang, L. (eds) Computing and Combinatorics. COCOON 2002. Lecture Notes in Computer Science, vol 2387. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45655-4_59
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DOI: https://doi.org/10.1007/3-540-45655-4_59
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