When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.IPEC.2017.2
URN: urn:nbn:de:0030-drops-85690
URL: http://drops.dagstuhl.de/opus/volltexte/2018/8569/
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### Finding Small Weight Isomorphisms with Additional Constraints is Fixed-Parameter Tractable

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### Abstract

Lubiw showed that several variants of Graph Isomorphism are NP-complete, where the solutions are required to satisfy certain additional constraints [SICOMP 10, 1981]. One of these, called Isomorphism With Restrictions, is to decide for two given graphs X_1=(V,E_1) and X_2=(V,E_2) and a subset R\subseteq V\times V of forbidden pairs whether there is an isomorphism \pi from X_1 to X_2 such that i^\pi\ne j for all (i,j)\in R. We prove that this problem and several of its generalizations are in fact in \FPT: - The problem of deciding whether there is an isomorphism between two graphs that moves k vertices and satisfies Lubiw-style constraints is in FPT, with k and the size of R as parameters. The problem remains in FPT even if a conjunction of disjunctions of such constraints is allowed. As a consequence of the main result it follows that the problem to decide whether there is an isomorphism that moves exactly k vertices is in FPT. This solves a question left open in our article on exact weight automorphisms [STACS 2017]. - When the number of moved vertices is unrestricted, finding isomorphisms that satisfy a CNF of Lubiw-style constraints can be solved in FPT with access to a GI oracle. - Checking if there is an isomorphism &#960; between two graphs with complexity t is also in FPT with t as parameter, where the complexity of a permutation is the Cayley measure defined as the minimum number t such that \pi can be expressed as a product of t transpositions. - We consider a more general problem in which the vertex set of a graph X is partitioned into Red and Blue, and we are interested in an automorphism that stabilizes Red and Blue and moves exactly k vertices in Blue, where k is the parameter. This problem was introduced by [Downey and Fellows 1999], and we showed [STACS 2017] that it is W[1]-hard even with color classes of size 4 inside Red. Now, for color classes of size at most 3 inside Red, we show the problem is in FPT. In the non-parameterized setting, all these problems are NP-complete. Also, they all generalize in several ways the problem to decide whether there is an isomorphism between two graphs that moves at most k vertices, shown to be in FPT by Schweitzer [ESA 2011].

### BibTeX - Entry

@InProceedings{arvind_et_al:LIPIcs:2018:8569,
author =	{Vikraman Arvind and Johannes K{\"o}bler and Sebastian Kuhnert and Jacobo Tor{\'a}n},
title =	{{Finding Small Weight Isomorphisms with Additional Constraints is Fixed-Parameter Tractable}},
booktitle =	{12th International Symposium on Parameterized and Exact Computation (IPEC 2017)},
pages =	{2:1--2:13},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-051-4},
ISSN =	{1868-8969},
year =	{2018},
volume =	{89},
editor =	{Daniel Lokshtanov and Naomi Nishimura},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},