Counting edge-injective homomorphisms and matchings on restricted graph classes

We consider the $\#\mathsf{W}[1]$-hard problem of counting all matchings with exactly $k$ edges in a given input graph $G$; we prove that it remains $\#\mathsf{W}[1]$-hard on graphs $G$ that are line graphs or bipartite graphs with degree $2$ on one side. In our proofs, we use that $k$-matchings in line graphs can be equivalently viewed as edge-injective homomorphisms from the disjoint union of $k$ length-$2$ paths into (arbitrary) host graphs. Here, a homomorphism from $H$ to $G$ is edge-injective if it maps any two distinct edges of $H$ to distinct edges in $G$. We show that edge-injective homomorphisms from a pattern graph $H$ can be counted in polynomial time if $H$ has bounded vertex-cover number after removing isolated edges. For hereditary classes $\mathcal{H}$ of pattern graphs, we complement this result: If the graphs in $\mathcal{H}$ have unbounded vertex-cover number even after deleting isolated edges, then counting edge-injective homomorphisms with patterns from $\mathcal{H}$ is $\#\mathsf{W}[1]$-hard. Our proofs rely on an edge-colored variant of Holant problems and a delicate interpolation argument; both may be of independent interest.