The Parameterized Complexity of Fixing Number and Vertex Individualization in Graphs

TL;DR

This paper investigates the parameterized complexity of fixing number and individualization problems in graphs, revealing hardness results and efficient algorithms depending on parameters and graph properties.

Contribution

It establishes the parameterized complexity classifications for fixing and individualization problems, including FPT algorithms and hardness results, in various graph classes.

Findings

Both fixing number problems are MINI[1]-hard when parameterized by set size.

FPT algorithms exist for the dual problems when parameterized by the complement size.

For graphs with small color classes, individualization problems are polynomial-time solvable; for larger classes, they are W[P]-hard.

Abstract

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.