Identifying codes of cycles with odd orders

Abstract

The problem of the r-identifying code of a cycle C n has been solved totally when n is even. Recently, S. Gravier et al. give the r-identifying code for the cycle C n with the minimum cardinality for odd n, when n ≥ 3r + 2 and gcd(2r + 1, n) = 1. In this paper, we deal with the r-identifying code of the cycle C n for odd n, when n ≥ 3r + 2 and gcd(2r + 1, n) = 1.

Identifying Codes in Vertex-Transitive Graphs and Strongly Regular Graphs

“…Identifying codes have already been studied in different classes of vertex-transitive graphs, especially in cycles [6,21,28,39] and hypercubes [7,12,13,27,29]. In these examples, the order of the size of an optimal identifying code seems to always match its fractional value.…”

Optimal Identifying Codes in Cycles and Paths

“…If n = 2m(2r + 1) + 1 or n = (2m + 1)(2r + 1) + 2r for m ≥ 1, then M r (C n ) = (n + 1)/2 + 1, else M r (C n ) = (n + 1)/2 [11].…”

“…Identifying codes in many different kinds of underlying graphs have been examined (see [6]). Among them are cycles and paths [1,3,8,11]; see also [2,4,9].…”

“…We say that a code T ⊆ V is a transversal of G if for each edge e = uv ∈ E the vertex u or the vertex v belongs to T. A transversal is also sometimes called a vertex cover [10, p. 102] or an edge-covering set [11] of G. Definition 1.1 Let r be a positive integer. A code C ⊆ V is said to be r -identifying in G if for all u, v ∈ V the set I r (C; u) is nonempty and I r (C; u) = I r (C; v).…”

Optimal bounds on codes for location in circulant graphs

“…where the calculations are done modulo n. Previously, in [2,4,7,9,13,17,19,23], identifying and locating-dominating codes have been studied in the circulant graphs C n (1, 2, . .…”