Identifying codes for infinite triangular grids with a finite number of rows

Abstract

International audienceLet G T be the infinite triangular grid. For any positive integer k, we denote by T k the subgraph of G T induced by the vertex set {(x, y) ∈ Z × [k]}. A set C ⊂ V (G) is an identifying code in a graph G if for all v ∈ V (G), N [v] ∩ C = ∅, and for all u, v ∈ V (G), N [u]∩C = N [v]∩C, where N [x] denotes the closed neighborhood of x in G. The minimum density of an identifying code in G is denoted by d * (G). In this paper, we prove that d * (T 1) = d * (T 2) = 1/2, d * (T 3) = d * (T 4) = 1/3, d * (T 5) = 3/10, d * (T 6) = 1/3 and d * (T k) = 1/4 + 1/(4k) for every k ≥ 7 odd. Moreover, we prove that 1/4 + 1/(4k) ≤ d * (T k) ≤ 1/4 + 1/(2k) for every k ≥ 8 even

On the Minimum Size of an Identifying Code Over All Orientations of a Graph

“…This method was often use to get lower bounds on the size of identifying codes, in particular for infinite grids, see e.g. [2,6,7,8,9].…”

Optimal error-detecting open-locating-dominating set on the infinite triangular grid

“…Much study has been done concerning various graphical parameters and distinguishing sets on the TRI graph [1][2][3]7]. For instance, optimal constructions of OLD sets [3] and RED:OLD sets [7] on the TRI graph have already been explored.…”

“…Seo and Slater [7] previously found an upper bound of 5 9 for the DET:OLD set problem on the TRI graph. We have improved the bound by constructing a 36-vertex tile with 18 detectors which can be tessellated to form a DET:OLD set with density 1 2 . Figure 3 shows this particular solution; it is easy to verify that every vertex is at least 2-dominated and all pairs are 2 # -distinguished, thus satisfying the requirements for a DET:OLD set given in Theorem 11.…”

Minimum density of identifying codes of king grids

“…Particular interest was dedicated to grids as many processor networks have a grid topology. Many results have been obtained on square grids [4,1,9,2,11], triangular grids [12,10], and hexagonal grids [5,7,8]. In this paper, we study king grids, which are strong products of two paths.…”

“…They provided the tile depicted in Figure 1, which generates a periodic tiling of the plane with periods (0, 6) and (6, 0), yielding an identifying code C ∞ of the bidimensional infinite king grid with density 2 9 . In this paper, using the Discharging Method (see Section 3 of [10] for a detailed presentation of this technique for identifying codes), we provide the following tight general lower bound on the minimum density of identifying codes of king grids.…”