The Parameterized Complexity of Fixing Number and Vertex Individualization in Graphs

Abstract

In this paper we study the algorithmic complexity of the following problems: (1) Given a vertex-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 \(G\le \operatorname{Sym}(n) \) 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 \(\textsf {MINI[1]} \) -hard. For the dual problems, where \(k=n-|S|\) is the parameter, we give FPT algorithms. (2) A notion related to fixing is individualization, which is a useful technique combined with the Weisfeiler-Leman procedure in algorithms for Graph Isomorphism. We explore the complexity of individualization: the problem of computing the minimum number of vertices we need to individualize in a given graph such that color refinement results in a graph with useful structural properties in the context of Graph Isomorphism and the Weisfeiler-Leman procedure.