Counting and Finding Homomorphisms is Universal for Parameterized Complexity Theory

Counting homomorphisms from a graph $H$ into another graph $G$ is a fundamental problem of (parameterized) counting complexity theory. In this work, we study the case where \emph{both} graphs $H$ and $G$ stem from given classes of graphs: $H\in \mathcal{H}$ and $G\in \mathcal{G}$. By this, we combine the structurally restricted version of this problem, with the language-restricted version. Our main result is a construction based on Kneser graphs that associates every problem $\tt P$ in $\#\mathsf{W[1]}$ with two classes of graphs $\mathcal{H}$ and $\mathcal{G}$ such that the problem $\tt P$ is \emph{equivalent} to the problem $\#{\tt HOM}(\mathcal{H}\to \mathcal{G})$ of counting homomorphisms from a graph in $\mathcal{H}$ to a graph in $\mathcal{G}$. In view of Ladner's seminal work on the existence of $\mathsf{NP}$-intermediate problems [J.ACM'75] and its adaptations to the parameterized setting, a classification of the class $\#\mathsf{W[1]}$ in fixed-parameter tractable and $\#\mathsf{W[1]}$-complete cases is unlikely. Hence, obtaining a complete classification for the problem $\#{\tt HOM}(\mathcal{H}\to \mathcal{G})$ seems unlikely. Further, our proofs easily adapt to $\mathsf{W[1]}$. In search of complexity dichotomies, we hence turn to special graph classes. Those classes include line graphs, claw-free graphs, perfect graphs, and combinations thereof, and $F$-colorable graphs for fixed graphs $F$: If the class $\mathcal{G}$ is one of those classes and the class $\mathcal{H}$ is closed under taking minors, then we establish explicit criteria for the class $\mathcal{H}$ that partition the family of problems $\#{\tt HOM}(\mathcal{H}\to\mathcal{G})$ into polynomial-time solvable and $\#\mathsf{W[1]}$-hard cases. In particular, we can drop the condition of $\mathcal{H}$ being minor-closed for $F$-colorable graphs.