  In this paper we study the complexity of the following problems:
  Given a colored graph X=(V,E,c), compute a minimum cardinality set S of
vertices 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
on [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.
  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 (even in
logspace), while starting from color class size 4 they become W[P]-hard.
