A bijection theorem for domino tilings with diagonal impurities

We consider the dimer problem on a planar non-bipartite graph G, where there are two types of dimers one of which we regard as impurities. Computer simulations reveal a reminiscence of the Cheerios effect, that is, impurities are attracted to the boundary, which is the motivation to study this particular graph. Our main theorem is a variant of the Temperley bijection: a bijection between the set of dimer coverings and the set of spanning forests with certain conditions. We further discuss some implications of this theorem: (1) the local move connectedness yielding an ergodic Markov chain on the set of all possible dimer coverings, and (2) a rough bound for the number of dimer coverings and that for the probability of finding an impurity at a given edge, which is an extension of a result in (Nakano and Sadahiro in arXiv:0901.4824).