On robust cycle bases

Two types of robust cycle bases are defined via recursively nice arrangements; complete and bipartite complete graphs are shown to have such bases. It is shown that a diagram in a groupoid is commutative up to natural equivalence (cutne) if for each cycle in a robust basis of the graph underlying the diagram, the composition of the morphisms is naturally equivalent to the identity. For a hypercube Qn, it is shown that the commutativity (or cutne) of a particular subset of asymptotically 4n of the square faces forces commutativity (or cutne) of the entire diagram.