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Summary: Submitted exclusively to the London Mathematical Society
doi:10.1112/0000/000000
Base size, metric dimension and other invariants
of groups and graphs
Robert F. Bailey and Peter J. Cameron
Abstract
The base size of a permutation group, and the metric dimension of a graph, are two of a number
of related parameters of groups, graphs, coherent configurations and association schemes. They
have been repeatedly re-defined with different terminology in various different areas, including
computational group theory and the graph isomorphism problem. We survey results on these
parameters in their many incarnations, and propose a consistent terminology for them. We also
present some new results, including on the base sizes of wreath products in the product action,
and on the metric dimension of Johnson and Kneser graphs.
1. Introduction
The base size of a permutation group is the smallest number of points whose stabiliser is the
identity. The metric dimension of a graph is the smallest number of vertices such that all vertices
are uniquely determined by their distances to the chosen vertices. The two parameters are
related by a straightforward inequality: the metric dimension of a graph gives an upper bound
on the base size of its automorphism group. Neither parameter is new: base size has a history
dating back around 40 years, while metric dimension dates back over 30 years. Both parameters
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