On the metric dimension of imprimitive distance-regular graphs

TL;DR

This paper investigates the metric dimension of imprimitive distance-regular graphs, relating it to their halved and folded graphs, and provides precise results for specific infinite families like Taylor graphs.

Contribution

It extends the understanding of metric dimension from primitive to imprimitive distance-regular graphs, offering new bounds and exact results for certain families.

Findings

Relates metric dimension of imprimitive graphs to their halved and folded counterparts.

Provides precise metric dimension results for Taylor graphs and symmetric design incidence graphs.

Extends bounds on metric dimension beyond primitive 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, but also consider infinite families (including Taylor graphs and the incidence graphs of certain symmetric designs) where more precise results are possible.