  We discuss connecting tables with zero-one entries by a subset of a Markov
basis. In this paper, as a Markov basis we consider the Graver basis, which
corresponds to the unique minimal Markov basis for the Lawrence lifting of the
original configuration. Since the Graver basis tends to be large, it is of
interest to clarify conditions such that a subset of the Graver basis, in
particular a minimal Markov basis itself, connects tables with zero-one
entries. We give some theoretical results on the connectivity of tables with
zero-one entries. We also study some common models, where a minimal Markov
basis for tables without the zero-one restriction does not connect tables with
zero-one entries.
