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arXiv · 1802.01325

Optimal bounds on codes for location in circulant graphs

Ville Junnila·Tero Laihonen·Gabrielle Paris
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Abstract

Identifying and locating-dominating codes have been studied widely in circulant graphs of type $C_n(1,2,3,\dots, r)$ over the recent years. In 2013, Ghebleh and Niepel studied locating-dominating and identifying codes in the circulant graphs $C_n(1,d)$ 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 $C_n(1,d)$, $C_n(1,d-1,d)$ and $C_n(1,d-1,d,d+1)$. 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 $C_n(1,3)$ and $C_n(1,4).$

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Ville Junnila, Tero Laihonen, Gabrielle Paris. 2018-02-05. Optimal bounds on codes for location in circulant graphs. https://arxiv.org/abs/1802.01325

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