tailieunhanh - Báo cáo toán học: "Construction of Codes Identifying Sets of Vertices"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Construction of Codes Identifying Sets of Vertices. | Construction of Codes Identifying Sets of Vertices Sylvain Gravier CNRS - UJF ERTé Maths à Modeler Groupe de Recherche GeoD Laboratoire Leibniz 46 avenue Felix Viallet 38031 Grenoble Cedex France Julien Moncel CNRS - UJF ERTé Maths à Modeler Groupe de Recherche GeoD Laboratoire Leibniz 46 avenue Felix Viallet 38031 Grenoble Cedex France Submitted Feb 8 2005 Accepted Mar 1 2005 Published Mar 8 2005 Mathematics Subject Classifications 05C99 94B60 94C12 Abstract In this paper the problem of constructing graphs having a 1 -identifying code of small cardinality is addressed. It is known that the cardinality of such a code is bounded by Q log log nỳ. Here we construct graphs on n vertices having a 1 -identifying code of cardinality O 4 log n for all 2. We derive our construction from a connection between identifying codes and superimposed codes which we describe in this paper. 1 Codes identifying sets of vertices Let G V E be a simple non-oriented graph. For a vertex v 2 V let us denote by N v the closed neighborhood of v N v N v u v . Let C Q V be a subset of vertices of G and for all nonempty subset of at most vertices X Q V let us denote I X I X C N x n C. x2X If all the I X C s are distinct then we say that C separates the sets of at most vertices of G and if all the I X C s are nonempty then we say that C covers the sets of at most vertices of G. We say that C is a code identifying sets of at most vertices of G if and only if C covers and separates all the sets of at most vertices of G. The dedicated terminology 12 for such codes is 1 Ế -identifying codes. The sets I X are said to be the identifying sets of the corresponding X s. THE ELECTRONIC JOURNAL OF COMBINATORICS 12 2005 R13 1 Whereas C V is trivially always a code covering the sets of at most vertices of any graph G V E not every graph has a 1 -identifying code. For example if G contains two vertices u and v such that N u N v then G has no 1 -identifying code .