  We investigate the computational complexity of the graph primality testing
problem with respect to the direct product (also known as Kronecker, cardinal
or tensor product). In [1] Imrich proves that both primality testing and a
unique prime factorization can be determined in polynomial time for (finite)
connected and nonbipartite graphs. The author states as an open problem how
results on the direct product of nonbipartite, connected graphs extend to
bipartite connected graphs and to disconnected ones. In this paper we partially
answer this question by proving that the graph isomorphism problem is
polynomial-time many-one reducible to the graph compositeness testing problem
(the complement of the graph primality testing problem). As a consequence of
this result, we prove that the graph isomorphism problem is polynomial-time
Turing reducible to the primality testing problem. Our results show that
connectedness plays a crucial role in determining the computational complexity
of the graph primality testing problem.
