Abstract. We prove that there is a graph isomorphism test running in time [Formula: see text] on [Formula: see text]-vertex graphs excluding some [Formula: see text]-vertex graph as a minor. Previously known bounds were [Formula: see text] [I. N. Ponomarenko, J. Soviet Math., 55 (1991), pp. 1621–1643] and [Formula: see text] [L. Babai, Proceedings of the 48 th Annual ACM Symposium on Theory of Computing, 2016, pp. 684–697]. For the algorithm we combine recent advances in the group-theoretic graph isomorphism machinery with new graph-theoretic arguments.

We introduce the framework of Deep Weisfeiler Leman algorithms (DeepWL), which allows the design of purely combinatorial graph isomorphism tests that are more powerful than the well-known Weisfeiler-Leman algorithm. We prove that, as an abstract computational model, polynomial-time DeepWL-algorithms have exactly the same expressiveness as the logic Choiceless Polynomial Time (with counting) introduced by Blass, Gurevich, and Shelah (Ann. Pure Appl. Logic., 1999). It is a well-known open question whether the existence of a polynomial-time graph isomorphism test implies the existence of a polynomial-time canonisation algorithm. Our main technical result states that for each class of graphs (satisfying some mild closure condition), if there is a polynomial-time DeepWL isomorphism test, then there is a polynomial-time canonisation algorithm for this class. This implies that there is also a logic capturing polynomial time on this class.

We prove that there is a graph isomorphism test running in time npolylog(h)on n-vertex graphs excluding some h-vertex graph as a minor. Previously known bounds were npoly(h)(Ponomarenko, 1988) and npolylog(n)(Babai, STOC 2016). For the algorithm we combine recent advances in the group-theoretic graph isomorphism machinery with new graph-theoretic arguments.

We extend Babai's quasipolynomial-time graph isomorphism test (STOC 2016) and develop a quasipolynomial-time algorithm for the multiple-coset isomorphism problem. The algorithm for the multiple-coset isomorphism problem allows to exploit graph decompositions of the given input graphs within Babai's group-theoretic framework. We use it to develop a graph isomorphism test that runs in time $n^{\operatorname{polylog}(k)}$ where $n$ is the number of vertices and $k$ is the minimum treewidth of the given graphs and $\operatorname{polylog}(k)$ is some polynomial in $\operatorname{log}(k)$. Our result generalizes Babai's quasipolynomial-time graph isomorphism test.

We show that normalizers and permutational isomorphisms of permutation groups given by generating sets can be computed in time simply exponential in the degree of the groups. The result is obtained by exploiting canonical forms for permutation groups (up to permutational isomorphism).

We give a new FPT algorithm testing isomorphism of n -vertex graphs of tree-width k in time 2 kpolylog(k) n 3 , improving the FPT algorithm due to Lokshtanov, Pilipczuk, Pilipczuk, and Saurabh (FOCS 2014), which runs in time 2 O(k5 log k) n 5 . Based on an improved version of the isomorphism-invariant graph decomposition technique introduced by Lokshtanov et al., we prove restrictions on the structure of the automorphism groups of graphs of tree-width k . Our algorithm then makes heavy use of the group theoretic techniques introduced by Luks (JCSS 1982) in his isomorphism test for bounded degree graphs and Babai (STOC 2016) in his quasipolynomial isomorphism test. In fact, we even use Babai’s algorithm as a black box in one place. We also give a second algorithm that, at the price of a slightly worse running time 2 O(k2 log k) n 3 , avoids the use of Babai’s algorithm and, more importantly, has the additional benefit that it can also be used as a canonization algorithm.

We devise a unified framework for the design of canonization algorithms. Using hereditarily finite sets, we define a general notion of combinatorial objects that includes graphs, hypergraphs, relational structures, codes, permutation groups, tree decompositions, and so on.