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# The Parameterized Complexity of Fixing Number and Vertex Individualization in Graphs

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## Cite As

Vikraman Arvind, Frank Fuhlbrück, Johannes Köbler, Sebastian Kuhnert, and Gaurav Rattan. The Parameterized Complexity of Fixing Number and Vertex Individualization in Graphs. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 13:1-13:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.MFCS.2016.13

## 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.
##### Keywords
• parameterized complexity
• graph automorphism
• fixing number
• base size
• individualization

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## References

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