Finding the Fixing Number of Johnson Graphs J(n, k) for k Є {2; 3}

Abstract

The graph invariant, aptly named the fixing number, is the smallest number of vertices that, when fixed, eliminate all non-trivial automorphisms (or symmetries) of a graph. Although many graphs have established fixing numbers, Johnson graphs, a family of graphs related to the graph isomorphism problem, have only partially classified fixing numbers. By examining specific orbit sizes of the automorphism group of Johnson graphs and classifying the subsequent remaining subgroups of the automorphism group after iteratively fixing vertices, we provide exact minimal sequences of fixed vertices, in turn establishing the fixing number of infinitely many Johnson graphs. KEYWORDS: Graph Automorphism Groups; Symmetry Breaking; Fixing Number; Determining Number; Johnson Graphs; Kneser Graphs; Graph Invariants; Permutation Groups; Minimal Sized Bases.