Affiliations: Department of Mathematics and Statistics, University of Turku, FI-20014 Turku, Finland, [email protected]; [email protected]
Correspondence:
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Address for correspondence: Department of Mathematics and Statistics, University of Turku, FI-20014 Turku, Finland
Abstract: A subset S of vertices is a resolving set in a graph if every vertex has a unique array of distances to the vertices of S. Consequently, we can locate any vertex of the graph with the aid of the distance arrays. The problem of finding the smallest cardinality of a resolving set in a graph has been widely studied over the years. In this paper, we consider sets S which can locate several, say up to ℓ, vertices in a graph. These sets are called {ℓ}-resolving sets and the smallest cardinality of such a set is the {ℓ}-metric dimension of the graph. In this paper, we will give the {ℓ}-metric dimensions for trees and king grids. We will show that there are certain vertices that necessarily belong to an {ℓ}-resolving set. Moreover, we will classify all graphs whose {ℓ}-metric dimension equals ℓ.
Keywords: Resolving set, metric dimension, metric basis, locating several vertices, forced vertices, king grid
