The Parameterized Complexity of Fixing Number and Vertex Individualization in Graphs

In this paper we study the algorithmic complexity of following problems: (1) Given a vertex-colored graph X = (V,E,c) , compute minimum cardinality set vertices S⊆ V such that no nontrivial automorphism fixes all in S . A closely related problem is computing base for permutation group G ≤ Sym( n ) given by generators, i.e., subset ⊆ [ ] elements Our focus mainly on parameterized these problems. We show when k =| | treated as parameter, then both problems are MINI [1]-hard. For dual problems, where – give FPT algorithms. (2) notion to fixing individualization, which useful technique combined with Weisfeiler-Leman procedure algorithms Graph Isomorphism. explore individualization: number need individualize color refinement results structural properties context Isomorphism and procedure.