The Fine-Grained Complexity of Graph Homomorphism Parameterized by Clique-Width

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.