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=nS is the parameter, we give FPT~algorithms.
2. A notion closely related to fixing is called individualization. Individualization combined with the WeisfeilerLeman 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 = {13:113:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770163},
ISSN = {18688969},
year = {2016},
volume = {58},
editor = {Piotr Faliszewski and Anca Muscholl and Rolf Niedermeier},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2016/6429},
URN = {urn:nbn:de:0030drops64294},
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 