Let G be a connected graph. A vertex w strongly resolves a pair u, v of vertices of G if there exists some shortest u− w path containing v or some shortest v − w path containing u. A set W of vertices is a strong resolving set for G if every pair of vertices of G is strongly resolved by some vertex of W . The smallest cardinality of a strong resolving set for G is called the strong metric dimen...

The metric dimension of a graph G is the size of a smallest subset L ⊆ V (G) such that for any x, y ∈ V (G) there is a z ∈ L such that the graph distance between x and z differs from the graph distance between y and z. Even though this notion has been part of the literature for almost 40 years, the computational complexity of determining the metric dimension of a graph is still very unclear. Es...

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. Let {G1, G2, . . . , Gn} be a finite collection of graphs and each Gi has a fixed vertex v0i or a fixed edge e0i called a terminal vertex or edge, respectively. The vertex-amalgamation of G1, ...

We show that Metric Dimension on planar graphs is NP-complete.

In this short note, we observe that the problem of computing the strong metric dimension of a graph can be reduced to the problem of computing a minimum node cover of a transformed graph within an additive logarithmic factor. This implies both a 2-approximation algorithm and a (2−ε)-inapproximability for the problem of computing the strong metric dimension of a graph.

In this paper the metric dimension of (the incidence graphs of) particular partial linear spaces is considered. We prove that the metric dimension of an affine plane of order q ≥ 13 is 3q − 4 and describe all resolving sets of that size if q ≥ 23. The metric dimension of a biaffine plane (also called a flag-type elliptic semiplane) of order q ≥ 4 is shown to fall between 2q−2 and 3q−6, while fo...

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 dimension of cartesian products G H. We prove that the metric dimension of G G is tied in a strong sense to the minimum order of a so-called doubly resolving set ...

A set S of vertices in a graph G is a resolving set if for every pair of vertices u, v ∈ V (G) there is a vertex x ∈ S such that the distances d(x, v) 6= d(x, u). We define a new parameter res(G), the size of the smallest subset S of V (G) that is not a resolving set but every superset of S resolves G. We also demonstrate that for every triple (a, b, c), a ≤ (b + 1) ≤ c, there is a graph G in w...

In this paper we introduce a new notion of generalized metric, called i-metric. This generalization is made by changing the valuation space of the distance function. The result is an interesting distance function for the set of fuzzy numbers of Interval Type with non negative fuzzy numbers as values. This example of i-metric generates a topology in a very natural way, based on open balls. We pr...

Given a graph G, we say S ⊆ V (G) is resolving if for each pair of distinct u, v ∈ V (G) there is a vertex x in S where d(u, x) 6= d(v, x). The metric dimension of G is the minimum cardinality of all resolving sets. For w ∈ V (G), the distance from w to S, denoted d(w, S), is the minimum distance between w and the vertices of S. Given P = {P1, P2, . . . , Pk} an ordered partition of V (G) we sa...