On the Metric Dimension of Imprimitive Distance-Regular Graphs

Abstract

A resolving set for a graph $${\Gamma}$$ is a collection of vertices S, chosen so that for each vertex v, the list of distances from v to the members of S uniquely specifies v. The metric dimension of $${\Gamma}$$ is the smallest size of a resolving set for $${\Gamma}$$ . Much attention has been paid to the metric dimension of distance-regular graphs. Work of Babai from the early 1980s yields general bounds on the metric dimension of primitive distance-regular graphs in terms of their parameters. We show how the metric dimension of an imprimitive distance-regular graph can be related to that of its halved and folded graphs. We also consider infinite families (including Taylor graphs and the incidence graphs of certain symmetric designs) where more precise results are possible.