  Fisher established an explicit correspondence between the 2-dimensional Ising
model defined on a graph $G$ and the dimer model defined on a decorated version
$\GD$ of this graph \cite{Fisher}. In this paper we explicitly relate the dimer
model associated to the critical Ising model and critical cycle rooted spanning
forests (CRSFs). This relation is established through characteristic
polynomials, whose definition only depends on the respective fundamental
domains, and which encode the combinatorics of the model. We first show a
matrix-tree type theorem establishing that the dimer characteristic polynomial
counts CRSFs of the decorated fundamental domain $\GD_1$. Our main result
consists in explicitly constructing CRSFs of $\GD_1$ counted by the dimer
characteristic polynomial, from CRSFs of $G_1$ where edges are assigned
Kenyon's critical weight function \cite{Kenyon3}; thus proving a relation on
the level of configurations between two well known 2-dimensional critical
models.
