Abstract

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.

Highlights

  • Already early on in the beginning of research on the graph isomorphism problem a close connection to the structure and study of the automorphism group of a graph was observed

  • Mathon [11] argued that the isomorphism problem is polynomially equivalent to the task of computing a generating set for the automorphism group and to computing the size of the automorphism group

  • Babai’s quasipolynomial time algorithm for general graph isomorphism [1] adds several novel techniques to tame and manage the groups that may appear as the automorphism group of the input graphs

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Summary

Introduction

Already early on in the beginning of research on the graph isomorphism problem (which asks for structural equivalence of two given input graphs) a close connection to the structure and study of the automorphism group of a graph was observed. Lokshtanov, Pilipczuk, Pilipczuk, and Saurabh [9] designed a fixedparameter tractable isomorphism test for graphs of bounded tree width which has a running time of 2O(k5 log k) poly(n) This algorithm first “improves” a given input graph G to a graph Gk by adding an edge between every pair of vertices between which more than k pairwise internally vertex disjoint paths exist. Lokshtanov et al [9] show that, after fixing a vertex of sufficiently low degree, is it possible to compute an isomorphism-invariant tree decomposition whose bags have a size at most exponential in k and whose adhesion is at most O(k3) They use this invariant decomposition to compute a canonical form essentially by a brute-force dynamic programming algorithm. It is still open whether there is a graph canonization algorithm running in quasipolynomial time

Our Results
Preliminaries
Clique separators and improved graphs
Decomposing basic graphs
Coset-Hypergraph-Isomorphism
The isomorphism test
Canonization
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