Insufficiency of four known necessary conditions on string unavoidability

The reductive decision procedure for unavoidable strings was recently shown to have an exponential lower bound. Hence, as a special case of generalized pattern matching, the existence of an efficient algorithm deciding string unavoidability remains an interesting open question. It has been hypothesized that some combination of the four necessary conditions implied by the known decidability results would be sufficient. Three of these criteria are determined in polynomial time, and the fourth provides the needed recursion. In this paper, however, we demonstrate the existence of arbitrarily many avoidable strings meeting any extended conjunction of the four necessary conditions. These insufficiency results are achieved by analyzing the appropriate graphical interpretations of the given algorithms. We provide a new combinatorial operation on the corresponding strings and generate arbitrary counterexamples from an empirically located minimal set. Thus, string unavoidability cannot be efficiently decided by the known reductive method or its immediate implications.