On the Metric Dimension of Imprimitive Distance-Regular Graphs

Abstract

A resolving set for a graph Γ 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 Γ is the smallest size of a resolving set for Γ. 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 distanceregular 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.

On the metric dimension of incidence graphs

“…The results of Sections 2 and 3 show that graphs in both of these classes have metric dimension µ(Γ) = O( √ n log n) (where n is the number of vertices). This extends Babai's results [3,4], which give the same upper bound for primitive graphs (which form class AH1), and the results of [6], which do so for graphs in classes AH11-AH13. 1 The class AH7, which consists of the non-bipartite antipodal graphs of diameter 3, is now the only class remaining where no general bounds (other than the trivial ones) on the metric dimension are known.…”

“…In [6], the same bounds on µ(Γ) as found in Corollaries 2.6 and 2.7 were obtained, but in the special case that the design has a null polarity (an incidence-preserving bijection σ between points and blocks, where no point is incident to its image under σ). These results remove that additional assumption, and thus answering an open question from [6] in the affirmative.…”

“…In [6], the same bounds on µ(Γ) as found in Corollaries 2.6 and 2.7 were obtained, but in the special case that the design has a null polarity (an incidence-preserving bijection σ between points and blocks, where no point is incident to its image under σ). These results remove that additional assumption, and thus answering an open question from [6] in the affirmative. Corollary 2.6 also extends to the class of bipartite distance-regular graphs of diameter 3 a result of Babai [4], which asserts that µ(Γ) = O( √ n log n) for primitive distance-regular graphs on n vertices.…”

“…In [1, Theorem 2.9], Alfuraidan and Hall gave a result which groups distanceregular graphs into various classes, in terms of whether the graph is primitive, bipartite, antipodal (or both bipartite and antipodal), and its diameter. In [6], these classes were labelled AH1-AH13: class AH1 consists of the primitive graphs of diameter at least 2 and valency at least 3; classes AH2-AH4 consist of cycles, complete graphs and complete multipartite graphs respectively; class AH5 consists of the graphs obtained by deleting a perfect matching from a complete bipartite graph. The remaining classes consist of the remaining antipodal graphs; in the present paper, we are primarily concerned with the following two of them:…”

“…The converse is also true: any bipartite, antipodal distance-regular graph with diameter 4 is the incidence graph of a symmetric transversal design (see [14, §1.7]). Their halved graphs are complete multipartite, while their folded graphs are complete bipartite; consequently, the results of [6] using halving or folding to obtain bounds on metric dimension do not yield particularly useful results here. However, the methods of the previous section, using semi-resolving sets and split resolving sets, may be easily adapted to obtain better bounds on µ(Γ D ).…”