The Computational Complexity of Tutte Invariants for Planar Graphs

For each pair of algebraic numbers (x, y), the complexity of computing the Tutte polynomial T(G; x, y) of a planar graph G is determined. This computation is found to be (#P) over bar -complete except when ( x - 1)( y - 1) = 1, 2 or when ( x, y) is one of (1, 1), (- 1,- 1), ( j, j(2)), or ( j(2), j), where j = e(2 pi i/ 3), in which case it is polynomial time computable. A corollary gives the computational complexity of various enumeration problems for planar graphs.