Abstract
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 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 #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.
Notes
We need to assume here that the colors of the two identified edges agree.
That is, r is bounded by dk for some overall constant d.
Here, x is an indeterminate, so the quantity (7) is a polynomial in x.
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Acknowledgments
The authors thank Cornelius Brand and Markus Bläser for interesting discussions, and Johannes Schmitt for pointing out a proof of Lemma 20 and allowing us to use it in this paper.
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Most of this work was done while the authors were visiting the Simons Institute for the Theory of Computing. Radu Curticapean is supported by ERC grants PARAMTIGHT (No. 280152) and SYSTEMATICGRAPH (No. 725978) and VILLUM Foundation grant 16582.
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Curticapean, R., Dell, H. & Roth, M. Counting Edge-injective Homomorphisms and Matchings on Restricted Graph Classes. Theory Comput Syst 63, 987–1026 (2019). https://doi.org/10.1007/s00224-018-9893-y
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DOI: https://doi.org/10.1007/s00224-018-9893-y