Computational complexity of counting problems on 3-regular planar graphs

A variety of counting problems on 3-regular planar graphs are considered in this paper. We give a sufficient condition which guarantees that the coefficients of a homogeneous polynomial can be uniquely determined by its values on a recurrence sequence. This result enables us to use the polynomial interpolation technique in high dimension to prove the #P-completeness of problems on graphs with special requirements. Using this method, we show that #3-Regular Bipartite Planar Vertex Covers is #P-complete. Furthermore, we use Valiant’s Holant Theorem to construct a holographic reduction from it to #2,3-Regular Bipartite Planar Matchings, establishing the #P-completeness of the latter. Finally, we completely classify the problems #Planar Read-twice 3SAT with different ternary symmetric relations according to their computational complexity, by giving several more applications of holographic reduction in proving the #P-completeness of the corresponding counting problems.