Counting Edge-Injective Homomorphisms and Matchings on Restricted Graph Classes

We consider the parameterized problem of counting all matchings with exactly k edges in a given input graph G. This problem is #W [1-j-hard (Curticapean, ICALP 2013), so it is unlikely to admit f(k) . n(O(1)) time algorithms. We show that #W[1]-hardness persists even when the input graph G comes from restricted graph classes, such as line graphs and bipartite graphs of arbitrary constant girth and maximum degree two on one side. To prove the result for line graphs, we observe that k-matchings in line graphs can be equivalently viewed as edge-injective homomorphisms from the disjoint union of k paths of length two 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 It of pattern graphs, we obtain a full complexity dichotomy theorem by proving that counting edge-injective homomorphisms, restricted to patterns from It, is #W[1]-hard if no such bound exists. Our proofs rely on an edge-colored variant of Holant problems and a delicate interpolation argument; both may be of independent interest.