The generic homomorphism problem, which asks whether an input graph \(G\) admits a homomorphism into a fixed target graph \(H\), has been widely studied in the literature. In this article, we provide a fine-grained complexity classification of the running time of the homomorphism problem with respect to the clique-width of \(G\) (denoted \({\operatorname{cw}}\)) for virtually all choices of \(H\) under the Strong Exponential Time Hypothesis. In particular, we identify a property of \(H\) called the signature number \(s(H)\) and show that for each \(H\), the homomorphism problem can be solved in time \(\mathcal{O^{*}}(s(H)^{{\operatorname{cw}}})\). Crucially, we then show that this algorithm can be used to obtain essentially tight upper bounds. Specifically, we provide a reduction that yields matching lower bounds for each \(H\) that is either a projective core or a graph admitting a factorization with additional properties—allowing us to cover all possible target graphs under long-standing conjectures.

通用同态问题，即询问输入图 G 是否承认固定目标图 H 的同态，已在文献中得到广泛研究。在本文中，我们提供了同态问题的运行时间的细粒度复杂性分类，该分类涉及几乎所有强指数时间假设下\(H\)的选择。特别是，我们确定了 \(H\) 的一个称为签名数 \(s(H)\) 的属性，并表明对于每个 \(H\)，同态问题可以及时解决 \(\mathcal{O ^{*}}(s(H)^{{\operatorname{cw}}})\)。至关重要的是，我们随后证明该算法可用于获得本质上严格的上限。具体来说，我们提供了一种归约方法，可以为每个 \(H\) 生成匹配的下界，该下界要么是投影核心，要么是允许具有附加属性的因式分解的图，从而使我们能够覆盖长期猜想下的所有可能的目标图。