New Bounds for Codes Identifying Vertices in Graphs

Abstract

Let $G=(V,E)$ be an undirected graph. Let $C$ be a subset of vertices that we shall call a code. For any vertex $v\in V$, the neighbouring set $N(v,C)$ is the set of vertices of $C$ at distance at most one from $v$. We say that the code $C$ identifies the vertices of $G$ if the neighbouring sets $N(v,C), v\in V,$ are all nonempty and different. What is the smallest size of an identifying code $C$ ? We focus on the case when $G$ is the two-dimensional square lattice and improve previous upper and lower bounds on the minimum size of such a code.

Locating and identifying codes in circulant networks

“…In the definition of r-identifying codes and r-locating-dominating sets the neighbourhood N [u] is replaced by the set N r [u] = {x ∈ V : d(u, x) r} for a constant r 1, where d(u, x) is the graph distance between vertices u and x. The r-identifying and r-locating codes correspond to the identifying and locating codes in the rth power G r of G. Locating and identifying codes have received a great deal of attention from researchers [19,3,22,10,5,2,23]. In particular, locating and identifying codes in special classes of networks have been studied.…”

Exact Minimum Density of Codes Identifying Vertices in the Square Grid

“…Bounds on the density of ID codes in Z 2 were given in [5,6,16]. The best known upper bound is 7 20 .…”

“…It was shown in [5] and is given, e.g., by a configuration depicted in Figure 1, shifted by vectors (10a + b, 4b), a, b, ∈ Z. The best known lower bound was 15 43 (see [6] and [20]). Thus there was a gap of about 0.0012 between the upper and lower bound.…”

“…The minimum density of ID codes in Z 2 is 7 20 . Our approach is a development of the method suggested earlier in [6]. It relies on construction of a bipartite graph characterizing the relations between vertices in Z 2 and elements of C in their environments.…”

On Vertex Identifying Codes For Infinite Lattices

“…The square grid has been extensively studied. Bounds were given in [9], [11] and [27] before it was shown in [1] that the D(G S , 1) = 7/20.…”