# Counting Edge-injective Homomorphisms and Matchings on Restricted Graph Classes

@article{Curticapean2018CountingEH, title={Counting Edge-injective Homomorphisms and Matchings on Restricted Graph Classes}, author={Radu Curticapean and Holger Dell and Marc Roth}, journal={Theory of Computing Systems}, year={2018}, volume={63}, pages={987-1026} }

We consider the #W[1]-hard problem of counting all matchings with exactly k edges in a given input graph G; we prove that it remains #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…

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