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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(More)
Given a graph G and a subset W ⊆ V (G), a Steiner W-tree is a tree of minimum order that contains all of W. Let S(W) denote the set of all vertices in G that lie on some Steiner W-tree; we call S(W) the Steiner interval of W. If S(W) = V (G), then we call W a Steiner set of G. The minimum order of a Steiner set of G is called the Steiner number of G. Given(More)
Subject Classification: 05C12 (distance in graphs), 05C35 (extremal graph theory) Abstract A set of vertices S resolves a connected 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 G β,D be the set of graphs with metric dimension(More)
The Reverse Nearest Neighbor (RNN) problem is to find all points in a given data set whose nearest neighbor is a given query point. Given a set of blue points and a set of red points, the bichromatic version of the RNN problem, for a query blue point, is to find all the red points whose blue nearest neighbour is the given query point. In this paper, we(More)
Let R be a set of red points and B a set of blue points on the plane. In this paper we introduce a new concept, which we call coarseness, for measuring how blended the elements of S = R ∪ B are. For X ⊆ S, let ∇(X) = ||X ∩ R| − |X ∩ B|| be the bichromatic discrepancy of X. We say that a partition Π = {S 1 , S 2 ,. .. , S k } of S is convex if the convex(More)
The concept of (minimum) resolving set has proved to be useful and/or related to a variety of fields such as Chemistry [3,6], Robotic Navigation [5,8] and Combinatorial Search and Optimization [7]. This work is devoted to evaluating the so-called metric dimension of a finite connected graph, i.e., the minimum cardinality of a resolving set, for a number of(More)
The Reverse Nearest Neighbor (RNN) problem is to find all points in a given data set whose nearest neighbor is a given query point. Given a set of blue points and a set of red points, the bichromatic version of the RNN problem, for a query blue point, is to find all the red points whose blue nearest neighbour is the given query point. In this paper, we(More)
In this paper we study the separability in the plane by two criteria: double-wedge separability and Θ-separability. We give an O(N log N)-time optimal algorithm for computing all the vertices of separating double wedges of two disjoint sets of objects (points, segments, polygons and circles) and an O((N/Θ 0) log N)-time algorithm for computing a(More)
Let G be a finite simple connected graph. A vertex v is a boundary vertex of G if there exists a vertex u such that no neighbor of v is further away from u than v. We obtain a number of properties involving different types of boundary vertices: peripheral, contour and eccentric vertices. Before showing that one of the main results in [3] does not hold for(More)