Optimal bounds on codes for location in circulant graphs

Abstract

Identifying and locating-dominating codes have been studied widely in circulant graphs of type Cn(1,2,3,...,r) over the recent years. In 2013, Ghebleh and Niepel studied locating-dominating and identifying codes in the circulant graphs Cn(1,d) for d=3 and proposed as an open question the case of d>3. In this paper we study identifying, locating-dominating and self-identifying codes in the graphs Cn(1,d), Cn(1,d-1,d) and Cn(1,d-1,d,d+1). We give a new method to study lower bounds for these three codes in the circulant graphs using suitable grids. Moreover, we show that these bounds are attained for infinitely many parameters n and d. In addition, new approaches are provided which give the exact values for the optimal self-identifying codes in Cn(1,3) and Cn(1,4).