Optimal(r,≤3)-locating–dominating codes in the infinite king grid

On locating-dominating codes for locating large numbers of vertices in the infinite king grid

Assume that G = (V, E) is an undirected graph with vertex set V and edge set E. The ball Br(v) denotes the vertices within graphical distance r from v. A subset C ⊆ V is called an (r,≤ l)-locating-dominating code of type B if the sets Ir(F ) = ⋃ v∈F (Br(v)∩C) are distinct for all subsets F ⊆ V \C with at most l vertices. A subset C ⊆ V is an (r,≤ l)-locatingdominating code of type A if sets Ir(...

On Identifying Codes in the King Grid that are Robust Against Edge Deletions

Assume that G = (V,E) is an undirected graph, and C ⊆ V . For every v ∈ V , we denote Ir(G; v) = {u ∈ C : d(u, v) ≤ r}, where d(u, v) denotes the number of edges on any shortest path from u to v. If all the sets Ir(G; v) for v ∈ V are pairwise different, and none of them is the empty set, the code C is called r-identifying. If C is r-identifying in all graphs G that can be obtained from G by de...

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,≤ l)-identifying code if for all subsets X of vertices with size at most l, the sets {c ∈ C|∃u ∈ X, d(u, c) ≤ 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,≤ 2)-identifying...

On 2-edge robust r-identifying codes in the king grid

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

Given a graph G, an identifying code D ⊆ V (G) is a vertex set such that for any two distinct vertices v1, v2 ∈ V (G), the sets N [v1] ∩ D and N [v2] ∩ 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 ...