Locating and Identifying Codes in Circulant Networks

TL;DR

This paper determines the minimum sizes of locating and identifying codes in circulant networks C_n(1,3), providing exact formulas and bounds for these codes based on the network size.

Contribution

It introduces exact formulas for the smallest locating and identifying codes in circulant networks C_n(1,3), advancing understanding of code placement in these graphs.

Findings

Smallest locating code size is loor n/3 loor + c, with c rac{0,1}

Smallest identifying code size is loor 4n/11 loor + c', with c' rac{0,1}

Provides bounds and formulas for code sizes in circulant networks

Abstract

A set S of vertices of a graph G is a dominating set of G if every vertex u of G is either in S or it has a neighbour in S. In other words, S is dominating if the sets S\cap N[u] where u \in V(G) and N[u] denotes the closed neighbourhood of u in G, are all nonempty. A set S \subseteq V(G) is called a locating code in G, if the sets S \cap N[u] where u \in V(G) \setminus S are all nonempty and distinct. A set S \subseteq V(G) is called an identifying code in G, if the sets S\cap N[u] where u\in V(G) are all nonempty and distinct. We study locating and identifying codes in the circulant networks C_n(1,3). For an integer n>6, the graph C_n(1,3) has vertex set Z_n and edges xy where x,y \in Z_n and |x-y| \in {1,3}. We prove that a smallest locating code in C_n(1,3) has size \lceil n/3 \rceil + c, where c \in {0,1}, and a smallest identifying code in C_n(1,3) has size \lceil 4n/11 \rceil + c', where c' \in {0,1}.