Chip removal for computing the number of perfect matchings

We consider a transformation of a graph $G$ that replaces an induced subgraph $H$ of arbitrary size by a little new subgraph $h$. We choose $h$ in such a way that the equality $M(G)=xM(G')$ holds (where $G'$ is the new graph and the factor $x$ depends on the numbers of matchings of $H$ and its subgraphs). We describe how one can construct $h$ when $G$ is a planar graph and $H$ is a bipartite graph (with some restriction on the coloring of vertices connecting it with the other part of the graph $G$). For a planar bipartite graph $H$ with a small number of such vertices, we prove that the equality holds for an arbitrary graph $G$.