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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)

Let S be a point set in the plane in general position, such that its elements are partitioned into k classes or colors. In this paper we study several variants on problems related to the Erd˝ os-Szekeres theorem about subsets of S in convex position, when additional chromatic constraints are considered.

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)