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Báo cáo toán học: " Identifying codes with small radius in some infinite regular graphs"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Identifying codes with small radius in some infinite regular graphs. | Identifying codes with small radius in some infinite regular graphs Irene Charon Olivier Hudry and Antoine Lobstein Centre National de la Recherche Scientifique Ecole Nationale Superieure des Telecommunications 46 rue Barrault 75634 Paris Cedex 13 - France charon hudry lobsteing@infres.enst.fr Submitted October 10 2000 Accepted March 13 2002. MR Subject Classifications 05C70 68R10 94B65 Abstract Let G V E be a connected undirected graph and S a subset of vertices. If for all vertices v 2 V the sets Br v n S are all nonempty and different where Br v denotes the set of all points within distance r from v then we call S an r-identifying code. We give constructive upper bounds on the best possible density of r-identifying codes in four infinite regular graphs for small values of r. 1 Introduction Given a connected undirected graph G V E finite or infinite we define Br v the ball of radius r centred at a vertex v 2 V by Br v x 2 V d X v rg where d X v denotes the number of edges in any shortest path between v and X. Whenever d X v r we say that X and v r-cover each other or simply cover if there is no ambiguity . A set of vertices covers a vertex if at least one of its elements does. We call any nonempty subset S of V a code and its elements codewords. A code S is called r-identifying or identifying if the sets Br v n S v 2 V are all nonempty and different. The set Br v n S is called the r-identifying set or identifying set of v and will be denoted by ISr v or IS v . Two vertices which have different identifying sets are said to be r-separated or separated. Remark. For given graph G V E and integer r there exists an r-identifying code S c V if and only if Vvi W2 2 V vi v2 Br vi Br v . THE ELECTRONIC JOURNAL OF COMBINATORICS 9 2002 R11 1 0 0 Figure 1 The hexagonal grid part . Indeed if for all v1 v2 2 V Br v1 and Br v2 are different then S V is r-identifying. Conversely if for some v1 v2 2 V Br v1 Br v2 then for any code S Q V we have IS v1 IS v2 . For instance there is