  We consider the dimer problem on a non-bipartite graph $G$, where there are
two types of dimers one of which we regard impurities. Results of simulations
using Markov chain seem to indicate that impurities are tend to distribute on
the boundary, which we set as a conjecture. We first show that there is a
bijection between the set of dimer coverings on $G$ and the set of spanning
forests on two graphs which are made from $G$, with configuration of impurities
satisfying a pairing condition. This bijection can be regarded as a extension
of the Temperley bijection. We consider local move consisting of two
operations, and by using the bijection mentioned above, we prove local move
connectedness. We further obtained some bound of the number of dimer coverings
and the probability finding an impurity at given edge, by extending the
argument in our previous result.
