Sphere of influence graphs: Edge density and clique size

  • Authors:

  • T. S. Michael;T. Quint

  • Affiliations:

  • Mathematics Department, United States Naval Academy Annapolis, MD 21402, U.S.A.;Department of Operations Research, Yale University New Haven, CT 06520, U.S.A.

  • Venue:
  • Mathematical and Computer Modelling: An International Journal
  • Year:
  • 1994

Quantified Score

Hi-index 0.98

Visualization

Abstract

LetX={X"1,...,X"n} be a set of n points(n = 2) in the metric space M. Let r"i denote the minimum distance between X"i and any other point in X. The closed ball B@?"i with center X"i and radius r"i is the closed sphere of influence at X"i. The closed sphere of influence graph CSIG (M, X) has vertex set X with distinct vertices X"i and X"j adjacent provided B"i@?B"j@A. The graph G is an M-CSIG provided G is isomorphic to CSIG(M, X) for some set X of points in M. We prove that, for any metric space M, the clique number is bounded over the class of M-CSIGs if and only if there is a constant C@?, so that the inequality |E| @? C|V| holds whenever G = (V, E) is an M-CSIG. The proof uses Ramsey's Theorem. We also prove that if M = (R^d, @r) is a d-dimensional Minkowski space, then C @? 5^d - 32.