Abstract
Identifying and locating-dominating codes have been studied widely in circulant graphs. Recently, Ville Junnila et al. (Optimal bounds on codes for location in circulant graphs, Cryptography and Communications; 2019) studied identifying and locating-dominating codes in circulants C n 1 , d , C n 1 , d − 1 , d , and C n 1 , d − 1 , d , d + 1 . In this paper, identifying, locating, and self-identifying codes in the circulant graphs C n k , d , C n k , d − k , d , and C n k , d − k , d , d + k are studied, and this extends Junnila et al.’s results to general cases.
Highlights
By definition, it is easy to see every identifying code is a locating code and a self-identifying code is an identifying code
(ii) We have an infinite sequence of identifying codes in the circulant graphs Cn(k, d − k, d, d + k) such that their density tends to 2/9
In order to study the codes of circulant graphs, we first give the necessary and sufficient conditions of these circulants to be connected
Summary
E closed neighborhood of u ∈ V in Γ is denoted by NΓ[u] NΓ(u) ∪ {u}. A nonempty subset of vertices C⊆V is called code and its elements are called codewords, and the number of elements in it is called the order of C denoted by |C|. Niepel studied locating and identifying code numbers of Cn(1, 3) in [11], but they stated as an open question what happens in the graphs Cn(1, d) with d being greater than 3 and mentioned that the methods used in their paper seem to be nonapplicable. Remark 1 (i) ere is an infinite sequence of identifying codes in the circulant graphs Cn(k, d − k, d) such that their density tends to 1/4.
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