Soon after, Valiant found holographic algorithms with reductions to matchgates for #Pl-Rtw-Mon-3CNF and #7Pl-3/2Bip-VC.
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An important application of the cycle space are Whitney's planarity criterion and Mac Lane's planarity criterion, which give an algebraic characterization of the planar graphs.
The FKT algorithm, named after Fisher, Kasteleyn, and Temperley, counts the number of perfect matchings in a planar graph in polynomial time.
Applications of PQ trees include creating a contig map from DNA fragments, testing a matrix for the consecutive ones property, recognizing interval graphs and determining whether a graph is planar.
Heiko Harborth raised the question of whether every planar graph has a straight line representation in which all edge lengths are integers.
Planar separator theorem (graph theory) states that any planar graph can be split into smaller pieces by removing a small number of vertices.