Here we prove that counting maximum matchings in planar, bipartite graphs is #P-complete. This is somewhat surprising in the light that the number of perfect matchings in planar graphs can be computed in polynomial time. We also prove that counting non-necessarily perfect matchings in planar graphs is already #P-complete if the problem is restricted to bipartite graphs. So far hardness was proved only for general, non-necessarily bipartite graphs.

中文翻译：

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计算平面图中的最大匹配很困难

在这里，我们证明计算平面二部图中的最大匹配是#P-complete。鉴于可以在多项式时间内计算平面图中完美匹配的数量，这有点令人惊讶。我们还证明，如果问题仅限于二部图，则计算平面图中不必要的完美匹配已经是#P-complete。到目前为止，仅对一般的、不必要的二部图证明了硬度。