Let Γ=(V,E) be a graph and ‎W_(‎a)={w_1,…,w_k } be a subset of the vertices of Γ and v be a vertex of it. The k-vector r_2 (v∣ W_a)=(a_Γ (v,w_1),‎…‎ ,a_Γ (v,w_k)) is the adjacency representation of v with respect to W in which a_Γ (v,w_i )=min{2,d_Γ (v,w_i )} and d_Γ (v,w_i ) is the distance between v and w_i in Γ. W_a is called as an adjacency resolving set for Γ if distinct vertices of ...

Abstract. In this article, we define the transport dimension of probability measures on Rm using ramified optimal transportation theory. We show that the transport dimension of a probability measure is bounded above by the Minkowski dimension and below by the Hausdorff dimension of the measure. Moreover, we introduce a metric, called “the dimensional distance,” on the space of probability measu...

In [R.F. Bailey, K. Meagher, On the metric dimension of Grassmann graphs, arXiv:1010.4495 ], Bailey and Meagher obtained an upper bound on the metric dimension of Grassmann graphs. In this note we show that qn+d−1+⌊ d+1 n ⌋ is an upper bound on the metric dimension of bilinear forms graphs Hq(n, d)when n ≥ d ≥ 2. As a result, we obtain an improvement on Babai’s most general bound for the metric...

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 ...