Abstract
Combinatorics In this paper, we consider a concept of adaptive identification of vertices and sets of vertices in different graphs, which was recently introduced by Ben-Haim, Gravier, Lobstein and Moncel (2008). The motivation for adaptive identification comes from applications such as sensor networks and fault detection in multiprocessor systems. We present an optimal adaptive algorithm for identifying vertices in cycles. We also give efficient adaptive algorithms for identifying sets of vertices in different graphs such as cycles, king lattices and square lattices. Adaptive identification is also considered in Hamming spaces, which is one of the most widely studied graphs in the field of identifying codes.
Highlights
Let G = (V, E) be a simple connected undirected graph with V as the set of vertices and E as the set of edges
It is clearly possible that the values given by the first cr(G) − 1 queries of A are all equal to 0, i.e. that there are no faulty vertices in the r-balls of the first cr(G) − 1 queries
If all the queries output value 0, there clearly exist no faulty vertices in Tpk,q
Summary
The definition of identifying codes is based on the fact that all the queries have to be asked simultaneously. Let be the maximum number of faulty vertices in a graph G. The minimum cardinality of an (r, ≤ )identifying code in G is denoted by i(r,≤ )(G). The corresponding value is the minimum number of queries required in the worst case to identify the (at most ) faulty vertices and it is denoted by a(r,≤ )(G). In Ben-Haim et al [1], [2] and [3], adaptive (r, ≤ 1)-identification is considered in torii of square and king lattices They suggest that further study would be needed in these torii when > 1. We introduce a slightly modified version of adaptive (r, ≤ )-identification in Section 3 to enable the handling of larger , for example, in king lattices
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