Graph Isomorphism in Quasipolynomial Time Parameterized by Treewidth

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.Comment: 52 pages, 1 figur

An Improved Isomorphism Test for Bounded-Tree-Width Graphs

We give a new fpt algorithm testing isomorphism of n-vertex graphs of tree width k in time 2^{k polylog(k)} poly n, improving the fpt algorithm due to Lokshtanov, Pilipczuk, Pilipczuk, and Saurabh (FOCS 2014), which runs in time 2^{O(k^5 log k)}poly n. 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\u27s algorithm as a black box in one place. We give a second algorithm which, at the price of a slightly worse run time 2^{O(k^2 log k)}poly n, avoids the use of Babai\u27s algorithm and, more importantly, has the additional benefit that it can also be used as a canonization algorithm

Isomorphism Testing for Graphs Excluding Small Minors

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