scholarly journals Construction of Codes Identifying Sets of Vertices

10.37236/1910 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Sylvain Gravier ◽  
Julien Moncel
Keyword(s):  
Identifying Codes ◽  
Identifying Code ◽  
Superimposed Codes ◽  
Small Cardinality

In this paper the problem of constructing graphs having a $(1,\le \ell)$-identifying code of small cardinality is addressed. It is known that the cardinality of such a code is bounded by $\Omega\left({\ell^2\over\log \ell}\log n\right)$. Here we construct graphs on $n$ vertices having a $(1,\le \ell)$-identifying code of cardinality $O\left(\ell^4 \log n\right)$ for all $\ell \ge 2$. We derive our construction from a connection between identifying codes and superimposed codes, which we describe in this paper.

scholarly journals Lower Bounds for Identifying Codes in some Infinite Grids

10.37236/394 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Ryan Martin ◽  
Brendon Stanton
Keyword(s):  
Lower Bounds ◽  
Finite Graph ◽  
Infinite Graphs ◽  
Locally Finite ◽  
Identifying Codes ◽  
Identifying Code ◽  
Hexagonal Grids ◽  
Closed Neighborhood ◽  
Definition Of ◽  
Infinite Grids

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 unique. On 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 present new lower bounds for densities of codes for some small values of $r$ in both the square and hexagonal grids.

scholarly journals General Bounds for Identifying Codes in Some Infinite Regular Graphs

10.37236/1583 ◽  
2001 ◽  
Vol 8 (1) ◽  
Author(s):  
Irène Charon ◽  
Iiro Honkala ◽  
Olivier Hudry ◽  
Antoine Lobstein
Keyword(s):  
Undirected Graph ◽  
Upper Bounds ◽  
Regular Graphs ◽  
Lower And Upper Bounds ◽  
Identifying Codes ◽  
Identifying Code

Consider a connected undirected graph $G=(V,E)$ and a subset of vertices $C$. If for all vertices $v \in V$, the sets $B_r(v) \cap C$ are all nonempty and pairwise distinct, where $B_r(v)$ denotes the set of all points within distance $r$ from $v$, then we call $C$ an $r$-identifying code. We give general lower and upper bounds on the best possible density of $r$-identifying codes in three infinite regular graphs.

Minimum identifying codes in some graphs differing by matchings

2020 ◽  
Vol 12 (03) ◽  
pp. 2050046
Author(s):  
R. Nikandish ◽  
O. Khani Nasab ◽  
E. Dodonge
Keyword(s):  
Maximum Matching ◽  
Minimum Cardinality ◽  
Identifying Codes ◽  
Identifying Code ◽  
Positive Integers

For a vertex [Formula: see text] of a graph [Formula: see text], let [Formula: see text] be the set of [Formula: see text] with all of its neighbors in [Formula: see text]. A set [Formula: see text] of vertices is an identifying code of [Formula: see text] if the sets [Formula: see text] are nonempty and distinct for all vertices [Formula: see text] of [Formula: see text]. If [Formula: see text] admits an identifying code, then [Formula: see text] is called identifiable and the minimum cardinality of an identifying code of [Formula: see text] is denoted by [Formula: see text]. Let [Formula: see text] be two positive integers. In this paper, [Formula: see text] and [Formula: see text] are computed, where [Formula: see text] and [Formula: see text] represent the complement of a path and the complement of a cycle of order [Formula: see text], respectively. Among other results, [Formula: see text] is given, where [Formula: see text] is obtained from [Formula: see text] after deleting a maximum matching.

scholarly journals Identifying and Locating–Dominating Codes in (Random) Geometric Networks

2009 ◽  
Vol 18 (6) ◽  
pp. 925-952 ◽  
Author(s):  
T. MÜLLER ◽  
J.-S. SERENI
Keyword(s):  
Unit Disk ◽  
Random Graphs ◽  
Bipartite Graphs ◽  
Sharp Contrast ◽  
Minimum Size ◽  
Wireless Devices ◽  
Identifying Codes ◽  
Identifying Code ◽  
Unit Disk Graphs

We model a problem about networks built from wireless devices using identifying and locating–dominating codes in unit disk graphs. It is known that minimizing the size of an identifying code is -complete even for bipartite graphs. First, we improve this result by showing that the problem remains -complete for bipartite planar unit disk graphs. Then, we address the question of the existence of an identifying code for random unit disk graphs. We derive the probability that there exists an identifying code as a function of the radius of the disks, and we find that for all interesting ranges of r this probability is bounded away from one. The results obtained are in sharp contrast to those concerning random graphs in the Erdős–Rényi model. Another well-studied class of codes is that of locating–dominating codes, which are less demanding than identifying codes. A locating–dominating code always exists, but minimizing its size is still -complete in general. We extend this result to our setting by showing that this question remains -complete for arbitrary planar unit disk graphs. Finally, we study the minimum size of such a code in random unit disk graphs, and we prove that with probability tending to one, it is of size (n/r)2/3+o(1) if r ≤ /2−ϵ is chosen such that nr2 → ∞, and of size n1+o(1) if nr2 ≪ lnn.

scholarly journals Optimal Identifying Codes in Oriented Paths and Circuits

2021 ◽  
Vol 15 ◽  
pp. 9-14
Author(s):  
Ahmed Semri ◽  
Hillal Touati
Keyword(s):  
Sensor Networks ◽  
Connected Graph ◽  
Multiprocessor Systems ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Identifying Codes ◽  
Identifying Code ◽  
Faulty Processor ◽  
Classical Notion ◽  
Codes In Graphs

Identifying codes in graphs are related to the classical notion of dominating sets [1]. Since there first introduction in 1998 [2], they have been widely studied and extended to several application, such as: detection of faulty processor in multiprocessor systems, locating danger or threats in sensor networks. Let G=(V,E) an unoriented connected graph. The minimum identifying code in graphs is the smallest subset of vertices C, such that every vertex in V have a unique set of neighbors in C. In our work, we focus on finding minimum cardinality of an identifying code in oriented paths and circuits

scholarly journals Identifying Codes with Small Radius in Some Infinite Regular Graphs

10.37236/1628 ◽  
2002 ◽  
Vol 9 (1) ◽  
Author(s):  
Irène Charon ◽  
Olivier Hudry ◽  
Antoine Lobstein
Keyword(s):  
Undirected Graph ◽  
Upper Bounds ◽  
Small Radius ◽  
Regular Graphs ◽  
Identifying Codes ◽  
Identifying Code

Let $G=(V,E)$ be a connected undirected graph and $S$ a subset of vertices. If for all vertices $v \in V$, the sets $B_r(v) \cap S$ are all nonempty and different, where $B_r(v)$ denotes the set of all points within distance $r$ from $v$, then we call $S$ an $r$-identifying code. We give constructive upper bounds on the best possible density of $r$-identifying codes in four infinite regular graphs, for small values of $r$.

Identifying codes and watching systems in Kneser graphs

2017 ◽  
Vol 09 (01) ◽  
pp. 1750007 ◽  
Author(s):  
Maryam Roozbayani ◽  
Hamid Reza Maimani
Keyword(s):  
Minimum Size ◽  
Identifying Codes ◽  
Identifying Code ◽  
Closed Neighborhood ◽  
Kneser Graphs

Identifying code in graph [Formula: see text] is a subset [Formula: see text] of [Formula: see text] such that [Formula: see text] for every [Formula: see text] of [Formula: see text] and [Formula: see text] for every [Formula: see text] of [Formula: see text]. The minimum size of identifying codes of graph [Formula: see text] is denoted by [Formula: see text]. A watching system in a graph [Formula: see text] is a set [Formula: see text], where [Formula: see text] and [Formula: see text] is a subset of closed neighborhood of [Formula: see text] such that the sets [Formula: see text] are nonempty and distinct, for any [Formula: see text]. The minimum size of a watching system of [Formula: see text] is denoted by [Formula: see text]. In this paper, we show if [Formula: see text], then [Formula: see text] if [Formula: see text] (mod 3) and [Formula: see text] if [Formula: see text] (mod 3). Also we show that [Formula: see text]. This means that in this family of graphs the watching system is more efficient than identifying code.

r-Identifying codes in binary Hamming space, q-ary Lee space and incomplete hypercube

2019 ◽  
Vol 11 (02) ◽  
pp. 1950027
Author(s):  
R. Dhanalakshmi ◽  
C. Durairajan
Keyword(s):  
Lower Bounds ◽  
Upper Bounds ◽  
Identifying Codes ◽  
Identifying Code ◽  
Construction Techniques ◽  
Lee Metric ◽  
Hamming Space

We study about monotonicity of [Formula: see text]-identifying codes in binary Hamming space, q-ary Lee space and incomplete hypercube. Also, we give the lower bounds for [Formula: see text] where [Formula: see text] is the smallest cardinality among all [Formula: see text]-identifying codes in [Formula: see text] with respect to the Lee metric. We prove the existence of [Formula: see text]-identifying code in an incomplete hypercube. Also, we give the construction techniques for [Formula: see text]-identifying codes in the incomplete hypercubes in Secs. 4.1 and 4.2. Using these techniques, we give the tables (see Tables 1–6) of upper bounds for [Formula: see text] where [Formula: see text] is the smallest cardinality among all [Formula: see text]-identifying codes in an incomplete hypercube with [Formula: see text] processors. Also, we give the exact values of [Formula: see text] for small values of [Formula: see text] and [Formula: see text] (see Sec. 4.3).

scholarly journals Identifying Codes of Cartesian Product of Two Cliques of the Same Size

10.37236/879 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
S. Gravier ◽  
J. Moncel ◽  
A. Semri
Keyword(s):  
Cartesian Product ◽  
Minimum Cardinality ◽  
Identifying Codes ◽  
Identifying Code

We determine the minimum cardinality of an identifying code of $K_n\square K_n$, the Cartesian product of two cliques of same size. Moreover we show that this code is unique, up to row and column permutations, when $n\geq 5$ is odd. If $n\geq 4$ is even, we exhibit two distinct optimal identifying codes.

scholarly journals Locating-Dominating Sets and Identifying Codes in Graphs of Girth at least 5

10.37236/4562 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Camino Balbuena ◽  
Florent Foucaud ◽  
Adriana Hansberg
Keyword(s):  
Minimum Degree ◽  
Dominating Set ◽  
Upper Bounds ◽  
Disjoint Paths ◽  
Minimum Size ◽  
Dominating Sets ◽  
Identifying Codes ◽  
Identifying Code ◽  
Codes In Graphs ◽  
Vertex Disjoint

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 meeting these bounds.

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