Adaptive Identification in Torii in the King Lattice

Given a connected graph G = (V,E), Let r ≥ 1 be an integer and Br(v) denote the ball of radius r centered at v ∈ V , i.e., the set of all vertices within distance r from v. A subset of vertices C ⊆ V is an r-identifying code of G (for a given nonzero constant r ∈ N) if and only if all the sets Br(v) ∩ C are nonempty and pairwise distinct. These codes were introduced in [7] to model a fault-detection problem in multiprocessor systems. They are also used to devise location-detection schemes in the framework of wireless sensor networks. These codes enable one to locate a malfunctioning device in these networks, provided one scans all the vertices of the code. We study here an adaptive version of identifying codes, which enables to perform tests dynamically. The main feature of such codes is that they may require significantly fewer tests, compared to usual static identifying codes. In this paper we study adaptive identifying codes in torii in the king lattice. In this framework, adaptive identification can be closely related to a Rényi-type search problem studied by M. Ruszinkó [11]. Partially supported by ANR/NSC Project GraTel, ANR-09-blan-0373-01 and NSC99-2923-M-110001-MY3, 2010–2013 Partially supported by ANR Project IDEA, ANR-08-EMER-007, 2009–2011. the electronic journal of combinatorics 18 (2011), #P116 1 1 Notations Given a connected undirected graph G = (V,E) and an integer r ≥ 1, the r-ball centered at v ∈ V , denoted Br(v), is defined by Br(v) = {x ∈ V | d(x, v) ≤ r}, where d(x, v) denotes the number of edges in any shortest path between x and v. Whenever x ∈ Br(v), we say that x and v r-cover each other (or simply cover if there is no ambiguity). A set X ⊆ V is said to cover a set Y ⊆ V if every vertex in Y is covered by at least one vertex in X. A code C is a nonempty set of vertices, and its elements are called codewords. For each vertex v ∈ V , we denote by KC,r(v) = C ∩Br(v) the set of codewords which r-cover v. Two vertices v1 and v2 with KC,r(v1) 6= KC,r(v2) are said to be r-separated, or separated, by the code C. A code C such that |KC,r(v)| ≥ 1 for all v ∈ V is called an r-covering code of G (it is often also called an r-dominating set of G). In other words, the set of vertices V is r-covered by the set of r-balls centered at vertices of C. A code C such that |KC,r(v)| ≤ 1 for all v ∈ V is called an r-packing (of r-balls) in G. In other words, the r-balls centered at vertices of C are all pairwise disjoint. A code being both an r-covering code and an r-packing of G is called an r-perfect code. These notations are more or less standard [6]. A code C is called r-identifying (or simply identifying if there is no ambiguity), if the sets KC,r(v), v ∈ V , are all nonempty and distinct [7]. In other words, all vertices must be r-covered and pairwise r-separated by C. Notice that, for a given graph G = (V,E) and an integer r, there exists an r-identifying code C ⊆ V if and only if Br(v1) 6= Br(v2) holds for all v1, v2 ∈ V, v1 6= v2. In this case, we say that G is r-identifiable (or simply identifiable if there is no ambiguity). Obviously, not all graphs are identifiable. Structural properties of identifiable graphs are studied in [2]. In the following, the graphs we consider are all identifiable graphs. Let ir(G) denote the minimum cardinality of an r-identifying code of an r-identifiable graph G. Let us also denote cr(G) (resp. γr(G)) the maximum cardinality (resp. the minimum cardinality) of an r-packing of G (resp. of an r-covering code in G). Clearly, we always have cr(G) ≤ γr(G), with equality if and only if there exists an r-perfect code in G. the electronic journal of combinatorics 18 (2011), #P116 2