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On the SIG-dimension of trees under the L (a)-metric

Chandran, Sunil L and Chitnis, Rajesh and Kumar, Ramanjit (2013) On the SIG-dimension of trees under the L (a)-metric. In: Graphs and Combinatorics, 29 (4). pp. 773-794.

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Official URL: http://dx.doi.org/10.1007/s00373-012-1160-4

Abstract

Let where be a set of points in d-dimensional space with a given metric rho. For a point let r (p) be the distance of p with respect to rho from its nearest neighbor in Let B(p,r (p) ) be the open ball with respect to rho centered at p and having the radius r (p) . We define the sphere-of-influence graph (SIG) of as the intersection graph of the family of sets Given a graph G, a set of points in d-dimensional space with the metric rho is called a d-dimensional SIG-representation of G, if G is isomorphic to the SIG of It is known that the absence of isolated vertices is a necessary and sufficient condition for a graph to have a SIG-representation under the L (a)-metric in some space of finite dimension. The SIG-dimension under the L (a)-metric of a graph G without isolated vertices is defined to be the minimum positive integer d such that G has a d-dimensional SIG-representation under the L (a)-metric. It is denoted by SIG (a)(G). We study the SIG-dimension of trees under the L (a)-metric and almost completely answer an open problem posed by Michael and Quint (Discrete Appl Math 127:447-460, 2003). Let T be a tree with at least two vertices. For each let leaf-degree(v) denote the number of neighbors of v that are leaves. We define the maximum leaf-degree as leaf-degree(x). Let leaf-degree{(v) = alpha}. If |S| = 1, we define beta(T) = alpha(T) - 1. Otherwise define beta(T) = alpha(T). We show that for a tree where beta = beta (T), provided beta is not of the form 2 (k) - 1, for some positive integer k a parts per thousand yen 1. If beta = 2 (k) - 1, then We show that both values are possible.

Item Type: Journal Article
Additional Information: Copyright of this article belongs to Springer.
Keywords: Sphere of Influence Graphs; Trees; L-Infinity Norm; Intersection Graphs
Department/Centre: Division of Electrical Sciences > Computer Science & Automation
Depositing User: Francis Jayakanth
Date Deposited: 03 Sep 2013 06:32
Last Modified: 03 Sep 2013 06:32
URI: http://eprints.iisc.ac.in/id/eprint/46961

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