A quasicancellation property for the direct product of graphs

We are motivated by the following question concerning the direct product of graphs. If A × C ≅ B × C , what can be said about the relationship between A and B ? If cancellation fails, what properties must A and B share? We define a structural equivalence relation ∼ (called similarity) on graphs, weaker than isomorphism, for which A × C ≅ B × C implies A ∼ B . Thus cancellation holds, up to similarity. Moreover, if C is bipartite, then A × C ≅ B × C if and only if A ∼ B . We conjecture that the prime factorization of connected bipartite graphs is unique up to similarity of factors, and we offer some results supporting this conjecture.