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.

中文翻译：

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图的直接产品素性测试是 GI 难的

我们研究了关于直接积（也称为 Kronecker、基数或张量积）的图素性测试问题的计算复杂性。在 [1] 中，Imrich 证明，对于（有限）连通图和非二部图，素性检验和唯一素数分解都可以在多项式时间内确定。作者将非二部连通图的直接积的结果如何扩展到二部连通图和断开连接图作为一个开放问题。在本文中，我们通过证明图同构问题是多项式时间多一可归约到图复合性测试问题（图素性测试问题的补充）来部分回答这个问题。由于这个结果，我们证明了图同构问题是多项式时间图灵可归约到素性测试问题。我们的结果表明，连通性在确定图素性测试问题的计算复杂性方面起着至关重要的作用。