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The Parameterized Complexity of Fixing Number and Vertex Individualization in Graphs

Published:07 October 2022Publication History
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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 ≤ 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 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.

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        • Published in

          cover image ACM Transactions on Computation Theory
          ACM Transactions on Computation Theory  Volume 14, Issue 2
          June 2022
          89 pages
          ISSN:1942-3454
          EISSN:1942-3462
          DOI:10.1145/3551656
          Issue’s Table of Contents

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          Publication History

          • Published: 7 October 2022
          • Online AM: 2 September 2022
          • Accepted: 17 August 2022
          • Revised: 11 August 2022
          • Received: 5 February 2021
          Published in toct Volume 14, Issue 2

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