We give a fine-grained classification of evaluating the Tutte polynomial $T(G;x,y)$ on all integer points on graphs with small treewidth and cutwidth. Specifically, we show for any point $(x,y) \in \mathbb{Z}^2$ that either - can be computed in polynomial time, - can be computed in $2^{O(tw)}n^{O(1)}$ time, but not in $2^{o(ctw)}n^{O(1)}$ time assuming the Exponential Time Hypothesis (ETH), - can be computed in $2^{O(tw \log tw)}n^{O(1)}$ time, but not in $2^{o(ctw \log ctw)}n^{O(1)}$ time assuming the ETH, where we assume tree decompositions of treewidth $tw$ and cutwidth decompositions of cutwidth $ctw$ are given as input along with the input graph on $n$ vertices and point $(x,y)$. To obtain these results, we refine the existing reductions that were instrumental for the seminal dichotomy by Jaeger, Welsh and Vertigan~[Math. Proc. Cambridge Philos. Soc'90]. One of our technical contributions is a new rank bound of a matrix that indicates whether the union of two forests is a forest itself, which we use to show that the number of forests of a graph can be counted in $2^{O(tw)}n^{O(1)}$ time.

翻译：我们给出了在具有小树宽和割宽的图上对所有整数点评估Tutte多项式$T(G;x,y)$的细粒度分类。具体地，我们证明对于任意点$(x,y) \in \mathbb{Z}^2$，要么可在多项式时间内计算，要么可在$2^{O(tw)}n^{O(1)}$时间内计算但在指数时间假设（ETH）下不能在$2^{o(ctw)}n^{O(1)}$时间内计算，要么可在$2^{O(tw \log tw)}n^{O(1)}$时间内计算但在ETH下不能在$2^{o(ctw \log ctw)}n^{O(1)}$时间内计算，其中我们假设输入图（包含$n$个顶点）和点$(x,y)$的同时，还输入了树宽为$tw$的树分解和割宽为$ctw$的割宽分解。为获得这些结果，我们改进了Jaeger、Welsh和Vertigan~[Math. Proc. Cambridge Philos. Soc'90]开创性二分法中所使用的关键现有归约。我们的技术贡献之一是一个新矩阵的秩界，该矩阵用于判断两个森林的并集是否仍是森林，并利用它证明了图的森林数量可在$2^{O(tw)}n^{O(1)}$时间内计数。