In this paper we extend the context of Steiner ratio and examine the influence of Steiner points on the weight of a graph in two generalizations of the Euclidean minimum weight connected graph (MST). The studied generalizations are with respect to the weight function and the connectivity condition. First, we consider the Steiner ratio of Euclidean minimum weight connected graph under the budget...

1. Tue 26 July, 9-9:30 Marcia Fampa, COPPE, Federal University of Rio de Janeiro, Brazil Modeling the Euclidean Steiner Tree problem In the Euclidean Steiner Tree Problem, the goal is to find a network of minimum length interconnecting a set P of given points in the n-dimensional Euclidean space. Such networks may be represented by a tree T , where the set of nodes is given by the points in P ,...

A unit disk graph is the intersection graph of unit disks in the euclidean plane. We present a polynomial-time approximation scheme for the maximum weight independent set problem in unit disk graphs. In contrast to previously known approximation schemes, our approach does not require a geometric representation (specifying the coordinates of the disk centers). The approximation algorithm present...

Consider the following two graphs M a n d N, both with vertex set Z×Z, where Z is the set of all integers. In M, two vertices are adjacent when their euclidean distance is 1, while in N, adjacency is obtained when the distance is either 1 or 1/2. By definition, H is a metric subgraph of the graph G if the distance between any two points o f / 4 is the same as their distance in G. We determine a...

We consider unitary graphs attached to d n , for odd integer n, using an analogue of the Euclidean distance. These graphs are shown to be integral.

A computer code and algorithm are developed for the computer perception of molecular symmetry. The code generates and uses the Euclidian distance matrices of molecular structures to generate the permutationinversion group of the molecule. The permutation-inversion group is constructed as the automorphism group of the Euclidian distance matrix. Applications to several molecular structures and fu...

For any graph G with fixed boundary there exists a layout in the plane, which minimizes the maximum Euclidean distance of any node to its neighbors. This layout balances the length of the graph edges and is therefore called a (length-) balanced layout of G. Furthermore the existence of a unique optimal balanced layout L with the following properties is proved: (i) L is the minimal element of an...

We show that greedy geometric routing schemes exist for the Euclidean metric in R, for 3-connected planar graphs, with coordinates that can be represented succinctly, that is, with O(log n) bits, where n is the number of vertices in the graph.

A geometric t-spanner on a set of points in Euclidean space is a graph containing for every pair of points a path of length at most t times the Euclidean distance between the points. Informally, a spanner is O(k)-robust if deleting k vertices only harms O(k) other vertices. We show that on any one-dimensional set of n points, for any ε > 0, there exists an O(k)-robust 1-spanner with O(n1+ε) edg...

The advance of deep learning has shown great potential in applications (speech, image, and video classification). In these applications, models are trained by datasets Euclidean space with fixed dimensions sequences. Nonetheless, the rapidly increasing demands on analyzing non-Euclidean require additional research. Generally speaking, finding relationships elements representing such as weighted...