10 research outputs found
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
and a lower bound of . 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 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
Minimum density of identifying codes of king grids
International audienceA set C ⊆ V (G) is an identifying code in a graph G if for all v ∈ V (G), C[v] = ∅, and for all distinct u, v ∈ V (G), C[u] = C[v], where C[v] = N [v] ∩ C and N [v] denotes the closed neighbourhood of v in G. The minimum density of an identifying code in G is denoted by d * (G). In this paper, we study the density of king grids which are strong product of two paths. We show that for every king grid G, d * (G) ≥ 2/9. In addition, we show this bound is attained only for king grids which are strong products of two infinite paths. Given k ≥ 3, we denote by K k the (infinite) king strip with k rows. We prove that d * (K 3) = 1/3, d * (K 4) = 5/16, d * (K 5) = 4/15 and d * (K 6) = 5/18. We also prove that 2 9 + 8 81k ≤ d * (K k) ≤ 2 9 + 4 9k for every k ≥ 7
Locating and Identifying Codes in Circulant Networks
A set S of vertices of a graph G is a dominating set of G if every vertex u
of G is either in S or it has a neighbour in S. In other words, S is dominating
if the sets S\cap N[u] where u \in V(G) and N[u] denotes the closed
neighbourhood of u in G, are all nonempty. A set S \subseteq V(G) is called a
locating code in G, if the sets S \cap N[u] where u \in V(G) \setminus S are
all nonempty and distinct. A set S \subseteq V(G) is called an identifying code
in G, if the sets S\cap N[u] where u\in V(G) are all nonempty and distinct. We
study locating and identifying codes in the circulant networks C_n(1,3). For an
integer n>6, the graph C_n(1,3) has vertex set Z_n and edges xy where x,y \in
Z_n and |x-y| \in {1,3}. We prove that a smallest locating code in C_n(1,3) has
size \lceil n/3 \rceil + c, where c \in {0,1}, and a smallest identifying code
in C_n(1,3) has size \lceil 4n/11 \rceil + c', where c' \in {0,1}
An improved lower bound for (1,<=2)-identifying codes in the king grid
We call a subset of vertices of a graph a -identifying
code if for all subsets of vertices with size at most , the sets
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 -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 -identifying code of
the king grid has density at least 47/111
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 for H must have density at least 16/39 ≈ 0.410256. Both their upper and lower bounds were best known until now. Here we prove a lower bound of 12/29 ≈ 0.413793
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