When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.MFCS.2016.13
URN: urn:nbn:de:0030-drops-64294
URL: https://drops.dagstuhl.de/opus/volltexte/2016/6429/
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### The Parameterized Complexity of Fixing Number and Vertex Individualization in Graphs

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### Abstract

In this paper we study the complexity of the following problems:

1. Given a colored graph X=(V,E,c), compute a minimum cardinality set of vertices S (subset of V) such that no nontrivial automorphism of X fixes all vertices in S. A closely related problem is computing a minimum base S for a permutation group G <= S_n given by generators, i.e., a minimum cardinality subset S of [n] such that no nontrivial permutation in G fixes all elements of S. Our focus is mainly on the parameterized complexity of these problems. We show that when k=|S| is treated as parameter, then both problems are MINI[1]-hard. For the dual problems, where k=n-|S| is the parameter, we give FPT~algorithms.

2. A notion closely related to fixing is called individualization. Individualization combined with the Weisfeiler-Leman procedure is a fundamental technique in algorithms for Graph Isomorphism. Motivated by the power of individualization, in the present paper we explore the complexity of individualization: what is the minimum number of vertices we need to individualize in a given graph such that color refinement "succeeds" on it. Here "succeeds" could have different interpretations, and we consider the following: It could mean the individualized graph becomes: (a) discrete, (b) amenable, (c)compact, or (d) refinable. In particular, we study the parameterized versions of these problems where the parameter is the number of vertices individualized. We show a dichotomy: For graphs with color classes of size at most 3 these problems can be solved in polynomial time, while starting from color class size 4 they become W[P]-hard.

### BibTeX - Entry

```@InProceedings{arvind_et_al:LIPIcs:2016:6429,
author =	{Vikraman Arvind and Frank Fuhlbr{\"u}ck and Johannes K{\"o}bler and Sebastian Kuhnert and Gaurav Rattan},
title =	{{The Parameterized Complexity of Fixing Number and Vertex Individualization in Graphs}},
booktitle =	{41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)},
pages =	{13:1--13:14},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-016-3},
ISSN =	{1868-8969},
year =	{2016},
volume =	{58},
editor =	{Piotr Faliszewski and Anca Muscholl and Rolf Niedermeier},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/6429},
URN =		{urn:nbn:de:0030-drops-64294},
doi =		{10.4230/LIPIcs.MFCS.2016.13},
annote =	{Keywords: parameterized complexity, graph automorphism, fixing   number, base size, individualization}
}
```

 Keywords: parameterized complexity, graph automorphism, fixing number, base size, individualization Collection: 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016) Issue Date: 2016 Date of publication: 19.08.2016

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