Sphere coverings and identifying codes

  • Authors:

  • David Auger;Gérard Cohen;Sihem Mesnager

  • Affiliations:

  • INRIA Saclay and University of Paris XI, Paris, France;Telecom-Paristech, UMR 5141, CNRS, Paris, France;Department of Mathematics, University of Paris VIII and University of Paris XIII, CNRS UMR 7539 LAGA, Paris, France

  • Venue:
  • Designs, Codes and Cryptography
  • Year:
  • 2014

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Abstract

In any connected, undirected graph G = (V, E), the distance d(x, y) between two vertices x and y of G is the minimum number of edges in a path linking x to y in G. A sphere in G is a set of the form S r (x) = {y 驴 V : d(x, y) = r}, where x is a vertex and r is a nonnegative integer called the radius of the sphere. We first address in this paper the following question: What is the minimum number of spheres with fixed radius r 驴 0 required to cover all the vertices of a finite, connected, undirected graph G? We then turn our attention to the Hamming Hypercube of dimension n, and we show that the minimum number of spheres with any radii required to cover this graph is either n or n + 1, depending on the parity of n. We also relate the two above problems to other questions in combinatorics, in particular to identifying codes.