A New Lower Bound on the Density of Vertex Identifying Codes for the Infinite Hexagonal Grid

Given a graph G, an identifying code D subset of V(G) is a vertex set such that for any two distinct vertices v(1), v(2) is an element of V(G), the sets N[v(1)] boolean AND D and N[v(2)] boolean AND D are distinct and nonempty (here N[v] denotes a vertex v and its neighbors). We study the case when G is the infinite hexagonal grid H.Cohen et.al. constructed two identifying codes for H with density 3/7 and proved that any identifying code for H must have density at least 16/39 approximate to 0.410256. Both their upper and lower bounds were best known until now. Here we prove a lower bound of 12/29 approximate to 0.413793

Automated Discharging Arguments for Density Problems in Grids

Discharging arguments demonstrate a connection between local structure and global averages. This makes it an effective tool for proving lower bounds on the density of special sets in infinite grids. However, the minimum density of an identifying code in the hexagonal grid remains open, with an upper bound of $\frac{3}{7} \approx 0.428571$ and a lower bound of $\frac{5}{12}\approx 0.416666$. We present a new, experimental framework for producing discharging arguments using an algorithm. This algorithm replaces the lengthy case analysis of human-written discharging arguments with a linear program that produces the best possible lower bound using the specified set of discharging rules. We use this framework to present a lower bound of $\frac{23}{55} \approx 0.418181$ on the density of an identifying code in the hexagonal grid, and also find several sharp lower bounds for variations on identifying codes in the hexagonal, square, and triangular grids.Comment: This is an extended abstract, with 10 pages, 2 appendices, 5 tables, and 2 figure

An improved lower bound for (1,<=2)-identifying codes in the king grid

We call a subset $C$ of vertices of a graph $G$ a $(1,\leq \ell)$-identifying code if for all subsets $X$ of vertices with size at most $\ell$, the sets $\{c\in C |\exists u \in X, d(u,c)\leq 1\}$ are distinct. The concept of identifying codes was introduced in 1998 by Karpovsky, Chakrabarty and Levitin. Identifying codes have been studied in various grids. In particular, it has been shown that there exists a $(1,\leq 2)$-identifying code in the king grid with density 3/7 and that there are no such identifying codes with density smaller than 5/12. Using a suitable frame and a discharging procedure, we improve the lower bound by showing that any $(1,\leq 2)$-identifying code of the king grid has density at least 47/111

Improved Bounds for rr-Identifying Codes of the Hex Grid

For any positive integer $r$, an $r$-identifying code on a graph $G$ is a set $C\subset V(G)$ such that for every vertex in $V(G)$, the intersection of the radius-$r$ closed neighborhood with $C$ is nonempty and pairwise distinct. For a finite graph, the density of a code is $|C|/|V(G)|$, which naturally extends to a definition of density in certain infinite graphs which are locally finite. We find a code of density less than $5/(6r)$, which is sparser than the prior best construction which has density approximately $8/(9r)$.Comment: 12p

On Vertex Identifying Codes For Infinite Lattices

PhD Thesis--A compilation of the papers: "Lower Bounds for Identifying Codes in Some Infinite Grids", "Improved Bounds for r-identifying Codes of the Hex Grid", and "Vertex Identifying Codes for the n-dimensional Lattics" along with some other resultsComment: 91p

Optimal local identifying and local locating-dominating codes

We introduce two new classes of covering codes in graphs for every positive integer $r$. These new codes are called local $r$-identifying and local $r$-locating-dominating codes and they are derived from $r$-identifying and $r$-locating-dominating codes, respectively. We study the sizes of optimal local 1-identifying codes in binary hypercubes. We obtain lower and upper bounds that are asymptotically tight. Together the bounds show that the cost of changing covering codes into local 1-identifying codes is negligible. For some small $n$ optimal constructions are obtained. Moreover, the upper bound is obtained by a linear code construction. Also, we study the densities of optimal local 1-identifying codes and local 1-locating-dominating codes in the infinite square grid, the hexagonal grid, the triangular grid, and the king grid. We prove that seven out of eight of our constructions have optimal densities