Geodesic
Chip removal for computing the number of perfect matchings
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XXVI. Representation theory, dynamical systems, combinatorial methods, Tome 437 (2015), pp. 62-80 
O. V. Bursian. Chip removal for computing the number of perfect matchings. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XXVI. Representation theory, dynamical systems, combinatorial methods, Tome 437 (2015), pp. 62-80. http://geodesic.mathdoc.fr/item/ZNSL_2015_437_a3/
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     author = {O. V. Bursian},
     title = {Chip removal for computing the number of perfect matchings},
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     year = {2015},
     volume = {437},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2015_437_a3/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

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 $G$.

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