Color refinement is an important technique that works very well in practice for the graph isomorphism problem. Tinhofer graphs are the class of graphs for which refinement together with individualization correctly tests graph isomorphism against every other graph, irrespective of the choices of vertices made during individualization. Motivated by the fact that Tinhofer graphs form a natural boundary for efficient graph isomorphism tests based on color refinement, in this paper, we introduce a hierarchy of graph classes within the class of Tinhofer graphs. We call a graph $G$ $k$-Tinhofer if, after $k$ rounds of individualization and refinement, the resulting colored graphs remain isomorphic for every graph $H \cong G$, irrespective of the choices of vertices made during individualization. Arvind et al. (2017) studied a hierarchy of graph classes motivated by color refinement - discrete, amenable, Tinhofer, and refinable graphs. We show that the $k$-Tinhofer hierarchy lies between the class of all graphs and Tinhofer graphs, with refinable graphs coinciding with the first level of the hierarchy. We obtain two characterizations of $k$-Tinhofer graphs: an algebraic characterization in terms of orbit partitions induced by pointwise stabilizers of automorphism groups, and a combinatorial characterization in terms of individualization-refinement trees and quotient graphs. For every fixed integer $k \ge 0$, there exist vertex-colored graphs that are $k$-Tinhofer but not $(k + 1)$-Tinhofer. For every fixed integer $k \ge 0$, the problem of deciding whether a given $k$-Tinhofer graph is ($k + 1$)-Tinhofer is $P$-hard under uniform $\mathsf{AC^0}$ many-one reductions. We show that testing isomorphism between an $(n - k)$-Tinhofer graph $G$ and an arbitrary graph $H$ is fixed-parameter tractable with respect to the parameter $k$.

翻译：颜色精炼是一种在图同构问题中实践中表现极为出色的重要技术。廷霍夫图是指通过精炼与个体化相结合的方法，无论个体化过程中如何选择顶点，都能正确判定任意其他图是否与之同构的图类。鉴于廷霍夫图构成了基于颜色精炼的高效图同构测试的自然边界，本文在廷霍夫图类内部引入了一种图类的层次结构。我们称图 $G$ 为 $k$-廷霍夫图，如果经过 $k$ 轮个体化与精炼后，所得着色图对于每个同构于 $G$ 的图 $H$ 仍保持同构，无论个体化过程中如何选择顶点。Arvind等人（2017年）研究了由颜色精炼驱动的图类层次结构——离散图、适从图、廷霍夫图与可精炼图。我们证明 $k$-廷霍夫层次结构位于所有图类与廷霍夫图类之间，其中可精炼图恰好对应于该层次结构的第一层。我们得到了 $k$-廷霍夫图的两种刻画：一种是通过自同构群的逐点稳定子群诱导的轨道划分的代数刻画，另一种是通过个体化-精炼树与商图的组合刻画。对于每个固定整数 $k \ge 0$，存在顶点着色图是 $k$-廷霍夫图但并非 $(k+1)$-廷霍夫图。对于每个固定整数 $k \ge 0$，判定给定 $k$-廷霍夫图是否为 $(k+1)$-廷霍夫图的问题在均匀 $\mathsf{AC^0}$ 多一归约下是 $P$ 难的。我们证明，在参数 $k$ 下，判定 $(n-k)$-廷霍夫图 $G$ 与任意图 $H$ 是否同构是固定参数易解的。