The Parameterized Complexity of Fixing Number and Vertex Individualization in Graphs

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 ⊂ 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 ≤ Sn given by generators, i.e., a minimum cardinality subset S ⊂ [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 (even in logspace), while starting from color class size 4 they become W[P]-hard.