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.

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 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 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.

Average covering number for some graphs

2019 ◽  
Vol 53 (1) ◽  
pp. 261-268
Author(s):  
D. Doğan Durgun ◽  
Ali Bagatarhan
Keyword(s):  
Interconnection Networks ◽  
Undirected Graph ◽  
Cartesian Product ◽  
Covering Number ◽  
Minimum Cardinality ◽  
Covering Numbers ◽  
Covering Set ◽  
Local Covering

The interconnection networks are modeled by means of graphs to determine the reliability and vulnerability. There are lots of parameters that are used to determine vulnerability. The average covering number is one of them which is denoted by $ \overline{\beta }(G)$, where G is simple, connected and undirected graph of order n ≥ 2. In a graph G = (V(G), E(G)) a subset $ {S}_v\subseteq V(G)$ of vertices is called a cover set of G with respect to v or a local covering set of vertex v, if each edge of the graph is incident to at least one vertex of Sv. The local covering number with respect to v is the minimum cardinality of among the Sv sets and denoted by βv. The average covering number of a graph G is defined as β̅(G) = 1/|v(G)| ∑ν∈v(G)βν In this paper, the average covering numbers of kth power of a cycle $ {C}_n^k$ and Pn □ Pm, Pn □ Cm, cartesian product of Pn and Pm, cartesian product of Pn and Cm are given, respectively.

On the fixed-point type Sylvester matrix equations over complete commutative dioids

2014 ◽  
Vol 27 ◽  
Author(s):  
Benham Hashemi ◽  
Mahtab Mirzaei Khalilabadi ◽  
Hanieh Tavakolipour
Keyword(s):  
Fixed Point ◽  
Tensor Product ◽  
Cartesian Product ◽  
Matrix Equations ◽  
Minimum Cardinality ◽  
Sylvester Matrix ◽  
Product Graphs ◽  
Cartesian Product Graphs ◽  
Sylvester Matrix Equations ◽  
Point Type

This paper extends the concept of tropical tensor product defined by Butkovic and Fiedler to general idempotent dioids. Then, it proposes an algorithm in order to solve the fixed-point type Sylvester matrix equations of the form X = A ⊗ X ⊕ X ⊗ B ⊕ C. An application is discussed in efficiently solving the minimum cardinality path problem in Cartesian product graphs.

scholarly journals Identifying Codes of Lexicographic Product of Graphs

10.37236/2974 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Min Feng ◽  
Min Xu ◽  
Kaishun Wang
Keyword(s):  
Connected Graph ◽  
Necessary Condition ◽  
Lexicographic Product ◽  
Arbitrary Graph ◽  
Minimum Cardinality ◽  
Identifying Codes ◽  
Two Parameters ◽  
Sufficient And Necessary Condition ◽  
Product Of Graphs

Let $G$ be a connected graph and $H$ be an arbitrary graph. In this paper, we study the identifying codes of the lexicographic product $G[H]$ of $G$ and $H$. We first introduce two parameters of $H$, which are closely related to identifying codes of $H$. Then we provide the sufficient and necessary condition for $G[H]$ to be identifiable. Finally, if $G[H]$ is identifiable, we determine the minimum cardinality of identifying codes of $G[H]$ in terms of the order of $G$ and these two parameters of $H$.

Bounds for complete cototal domination number of Cartesian product graphs and complement graphs

2021 ◽  
pp. 2150090
Author(s):  
J. Maria Regila Baby ◽  
K. Uma Samundesvari
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Cartesian Product ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Product Graphs ◽  
Cartesian Product Graphs

A total dominating set [Formula: see text] is said to be a complete cototal dominating set if [Formula: see text] has no isolated nodes and it is represented by [Formula: see text]. The complete cototal domination number, represented by [Formula: see text], is the minimum cardinality of a [Formula: see text] set of [Formula: see text]. In this paper, the bounds for complete cototal domination number of Cartesian product graphs and complement graphs are determined.

On identifying codes in the Cartesian product of a path and a complete graph

2015 ◽  
Vol 31 (4) ◽  
pp. 1405-1416 ◽  
Author(s):  
Jason Hedetniemi
Keyword(s):  
Complete Graph ◽  
Cartesian Product ◽  
Identifying Codes

scholarly journals Monotonicity of the minimum cardinality of an identifying code in the hypercube

2006 ◽  
Vol 154 (6) ◽  
pp. 898-899 ◽  
Author(s):  
Julien Moncel
Keyword(s):  
Minimum Cardinality ◽  
Identifying Code
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