The problem of computing the Tutte polynomial of a graph has been a hot topic in recent years,
because its computation is very useful not only in graph theory but also in many problems in statistical
physics, knot theory, etc. This problem is #P-hard, and there have been known only algorithms taking
time at least proportional to the number of trees of a given graph which is exponential in nature. This
paper presents a new algorithm by utilizing a fact that many 2-isomorphic minors appear in the process
of computation. The complexity of the algorithm is analyzed in terms of Bell numbers and Catalan
numbers in combinatorics. By this algorithm, we can compute the Tutte polynomial of any graph with
at most 14 vertices and 91 edges and that of a planar graph such as 12212 lattice graph with 144 vertices
and 264 edges by a standard workstation in about an hour.