On the Metric Dimensions for Sets of Vertices

Abstract

Resolving sets were originally designed to locate vertices of a graph one at a time. For the purpose of locating multiple vertices of the graph simultaneously, $\{\ell \}$-resolving sets were recently introduced. In this paper, we present new results regarding the $\{\ell \}$-resolving sets of a graph. In addition to proving general results, we consider $\{2\}$-resolving sets in rook's graphs and connect them to block designs. We also introduce the concept of $\ell$-solid-resolving sets, which is a natural generalisation of solid-resolving sets. We prove some general bounds and characterisations for $\ell$-solid-resolving sets and show how $\ell$-solid- and $\{\ell \}$-resolving sets are connected to each other. In the last part of the paper, we focus on the infinite graph family of flower snarks. We consider the $\ell$-solid- and $\{\ell \}$-metric dimensions of flower snarks. In two proofs regarding flower snarks, we use a new computer-aided reduction-like approach.

21 pages, 5 figures

Cited by in corpus (2)

• On the unicyclic graphs having vertices that belong to all their (strong) metric bases
• On Vertices Contained in All or in No Metric Basis