18,135 research outputs found
Minimum-Density Identifying Codes in Square Grids
International audienceAn identifying code in a graph G is a subset of vertices with the property that for each vertex v ∈ V (G), the collection of elements of C at distance at most 1 from v is non-empty and distinct from the collection of any other vertex. We consider the minimum density d * (S k) of an identifying code in the square grid S k of height k (i.e. with vertex set Z × {1,. .. , k}). Using the Discharging Method, we prove 7 20 + 1 20k ≤ d * (S k) ≤ min 2 5 , 7 20 + 3 10k , and d * (S3) = 7 18
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
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
Improved Bounds for -Identifying Codes of the Hex Grid
For any positive integer , an -identifying code on a graph is a set
such that for every vertex in , the intersection of the
radius- closed neighborhood with is nonempty and pairwise distinct. For
a finite graph, the density of a code is , which naturally extends
to a definition of density in certain infinite graphs which are locally finite.
We find a code of density less than , which is sparser than the prior
best construction which has density approximately .Comment: 12p
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