Perfect Matchings in a Planar Graphs

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Perfect Matchings in a Planar Graphs

Perfect Matchings in a Planar Graphs

Thm 1.11 (Kasteleyn):

For any Pfaffian orientation G → ,

# perfect matchings of G = det A s (G → )

Thm 1.14: Every planar graph has a Pfaffian orientation

(and it can be easily constructed).

[Section 1.2]

Note: It is not known whether the problem of deciding if a

graph has a Pfaffian orientation is in P nor whether it is NPcomplete.

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