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The metric dimension of a graph is the smallest number of nodes required to identify all other nodes uniquely based on shortest path distances. Applications of metric dimension include discovering the source of a spread in a network, canonically labeling ...

A subset of vertices in a graph is called resolving when the geodesic distances to those vertices uniquely distinguish every vertex in the graph. Here, we characterize the resolvability of Hamming graphs in terms of a constrained linear system and deduce ...

The symbolic nature of biological sequence data greatly complicates its analysis. As many of the most powerful, widely used analysis techniques work exclusively in the context of Euclidean spaces, methods of embedding DNA, RNA, and protein sequences in ...

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 ...

The metric dimension of a graph is the minimum size of a set of vertices such that each vertex is uniquely determined by the distances to the vertices of that set. Our aim is to upper-bound the order $n$ of a graph in terms of its diameter $d$ and metric ...

We introduce the notion of a centroidal locating set of a graph G, that is, a set L of vertices such that all vertices in G are uniquely determined by their relative distances to the vertices of L. A centroidal locating set of G of minimum size is ...

Let (W,d) be a metric space and S={s₁ …s_k} an ordered list of subsets of W. The distance between p∈W and s_i∈S is d(p, s_i)= min { d(p,q) : q∈s_i }. S is a resolving set for W if d(x, s_i)=d(y, s_i) for all s_i implies x=y. A metric basis is a resolving set ...

Let (W, d) be a metric space. A subset S ⊆ W is a resolving set for W if d(x, p) = d(y, p) for all p ∈ S implies x = y. A metric basis is a resolving set of minimal cardinality, named the metric dimension (of W). Metric bases and dimensions have been ...

We study the approximation complexity of the Metric Dimension problem in bounded degree, dense as well as in general graphs. For the general case, we prove that the Metric Dimension problem is not approximable within (1-ε)lnn for any ε>0, unless NP⊆...

A set of vertices $S$ resolves a graph $G$ if every vertex is uniquely determined by its vector of distances to the vertices in $S$. The metric dimension of $G$ is the minimum cardinality of a resolving set of $G$. This paper studies the metric ...