9,590 research outputs found
Fault-Tolerant Detection Systems on the King's Grid
A detection system, modeled in a graph, uses "detectors" on a subset of
vertices to uniquely identify an "intruder" at any vertex. We consider two
types of detection systems: open-locating-dominating (OLD) sets and identifying
codes (ICs). An OLD set gives each vertex a unique, non-empty open neighborhood
of detectors, while an IC provides a unique, non-empty closed neighborhood of
detectors. We explore their fault-tolerant variants: redundant OLD (RED:OLD)
sets and redundant ICs (RED:ICs), which ensure that removing/disabling at most
one detector guarantees the properties of OLD sets and ICs, respectively. This
paper focuses on constructing optimal RED:OLD sets and RED:ICs on the infinite
king's grid, and presents the proof for the bounds on their minimum densities;
[3/10, 1/3] for RED:OLD sets and [3/11, 1/3] for RED:ICs
Locating-dominating sets and identifying codes in graphs of girth at least 5
Locating-dominating sets and identifying codes are two closely related
notions in the area of separating systems. Roughly speaking, they consist in a
dominating set of a graph such that every vertex is uniquely identified by its
neighbourhood within the dominating set. In this paper, we study the size of a
smallest locating-dominating set or identifying code for graphs of girth at
least 5 and of given minimum degree. We use the technique of vertex-disjoint
paths to provide upper bounds on the minimum size of such sets, and construct
graphs who come close to meet these bounds.Comment: 20 pages, 9 figure
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
Optimal Index Codes via a Duality between Index Coding and Network Coding
In Index Coding, the goal is to use a broadcast channel as efficiently as
possible to communicate information from a source to multiple receivers which
can possess some of the information symbols at the source as side-information.
In this work, we present a duality relationship between index coding (IC) and
multiple-unicast network coding (NC). It is known that the IC problem can be
represented using a side-information graph (with number of vertices
equal to the number of source symbols). The size of the maximum acyclic induced
subgraph, denoted by is a lower bound on the \textit{broadcast rate}.
For IC problems with and , prior work has shown that
binary (over ) linear index codes achieve the lower bound
for the broadcast rate and thus are optimal. In this work, we use the the
duality relationship between NC and IC to show that for a class of IC problems
with , binary linear index codes achieve the lower bound on
the broadcast rate. In contrast, it is known that there exists IC problems with
and optimal broadcast rate strictly greater than
Vertex identifying codes for the n-dimensional lattice
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 different. Here, we provide an overview on codes for
the -dimensional lattice, discussing the case of 1-identifying codes,
constructing a sparse code for the 4-dimensional lattice as well as showing
that for fixed , the minimum density of an -identifying code is
.Comment: 10p
Centroidal bases in graphs
We introduce the notion of a centroidal locating set of a graph , that is,
a set of vertices such that all vertices in are uniquely determined by
their relative distances to the vertices of . A centroidal locating set of
of minimum size is called a centroidal basis, and its size is the
centroidal dimension . This notion, which is related to previous
concepts, gives a new way of identifying the vertices of a graph. The
centroidal dimension of a graph is lower- and upper-bounded by the metric
dimension and twice the location-domination number of , respectively. The
latter two parameters are standard and well-studied notions in the field of
graph identification.
We show that for any graph with vertices and maximum degree at
least~2, . We discuss the
tightness of these bounds and in particular, we characterize the set of graphs
reaching the upper bound. We then show that for graphs in which every pair of
vertices is connected via a bounded number of paths,
, the bound being tight for paths and
cycles. We finally investigate the computational complexity of determining
for an input graph , showing that the problem is hard and cannot
even be approximated efficiently up to a factor of . We also give an
-approximation algorithm
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