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On the Complexity of L-reachability

We initiate a complexity theoretic study of the language based reachability problem (L-reach) : Fix a language L. Given a graph whose edges are labelled with alphabet symbols and two special vertices s and t, test if there is path P from s to t in the graph such that the concatenation of the symbols seen from s to t in the path P forms a string in the language L. We study variants of this problem with different graph classes and different language classes and obtain complexity theoretic characterisations for all of them. Our main results are the following: