A Hierarchy of Tinhofer Graphs: Separations and Membership Testing

2026-05-19

AI Summary

This work investigates the expressive power of color-refinement-based graph isomorphism testing within the class of Tinhofer graphs. To this end, it introduces the hierarchy of $k$-Tinhofer graphs, which characterizes those graphs whose isomorphism-invariance persists after $k$ rounds of individualization and refinement. Employing tools from group theory—such as point stabilizer orbit partitions, individualization-refinement trees, and quotient graph constructions—the paper establishes that this hierarchy strictly lies between the class of all graphs and the class of Tinhofer graphs: for any fixed $k$, there exist graphs that are $k$-Tinhofer but not $(k+1)$-Tinhofer; graph isomorphism for $(n-k)$-Tinhofer graphs is fixed-parameter tractable in $k$; and the membership problem for $k$-Tinhofer graphs is P-hard.

Abstract

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 $k$.

Problem

• Tinhofer graphs
• graph isomorphism
• color refinement
• individualization
• hierarchy

Innovation

• k-Tinhofer graphs
• color refinement
• individualization-refinement
• graph isomorphism
• fixed-parameter tractability