Counting edgeinjective homomorphisms and matchings on restricted graph classes
Abstract
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 edgeinjective homomorphisms from the disjoint union of $k$ length$2$ paths into (arbitrary) host graphs. Here, a homomorphism from $H$ to $G$ is edgeinjective if it maps any two distinct edges of $H$ to distinct edges in $G$. We show that edgeinjective homomorphisms from a pattern graph $H$ can be counted in polynomial time if $H$ has bounded vertexcover 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 vertexcover number even after deleting isolated edges, then counting edgeinjective homomorphisms with patterns from $\mathcal{H}$ is $\#\mathsf{W}[1]$hard. Our proofs rely on an edgecolored variant of Holant problems and a delicate interpolation argument; both may be of independent interest.
 Publication:

arXiv eprints
 Pub Date:
 February 2017
 arXiv:
 arXiv:1702.05447
 Bibcode:
 2017arXiv170205447C
 Keywords:

 Computer Science  Computational Complexity
 EPrint:
 35 pages, 9 figures