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 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.

scholarly journals Distance Domination and Distance Irredundance in Graphs

10.37236/953 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Adriana Hansberg ◽  
Dirk Meierling ◽  
Lutz Volkmann
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Distance Domination

A set $D\subseteq V$ of vertices is said to be a (connected) distance $k$-dominating set of $G$ if the distance between each vertex $u\in V-D$ and $D$ is at most $k$ (and $D$ induces a connected graph in $G$). The minimum cardinality of a (connected) distance $k$-dominating set in $G$ is the (connected) distance $k$-domination number of $G$, denoted by $\gamma_k(G)$ ($\gamma_k^c(G)$, respectively). The set $D$ is defined to be a total $k$-dominating set of $G$ if every vertex in $V$ is within distance $k$ from some vertex of $D$ other than itself. The minimum cardinality among all total $k$-dominating sets of $G$ is called the total $k$-domination number of $G$ and is denoted by $\gamma_k^t(G)$. For $x\in X\subseteq V$, if $N^k[x]-N^k[X-x]\neq\emptyset$, the vertex $x$ is said to be $k$-irredundant in $X$. A set $X$ containing only $k$-irredundant vertices is called $k$-irredundant. The $k$-irredundance number of $G$, denoted by $ir_k(G)$, is the minimum cardinality taken over all maximal $k$-irredundant sets of vertices of $G$. In this paper we establish lower bounds for the distance $k$-irredundance number of graphs and trees. More precisely, we prove that ${5k+1\over 2}ir_k(G)\geq \gamma_k^c(G)+2k$ for each connected graph $G$ and $(2k+1)ir_k(T)\geq\gamma_k^c(T)+2k\geq |V|+2k-kn_1(T)$ for each tree $T=(V,E)$ with $n_1(T)$ leaves. A class of examples shows that the latter bound is sharp. The second inequality generalizes a result of Meierling and Volkmann and Cyman, Lemańska and Raczek regarding $\gamma_k$ and the first generalizes a result of Favaron and Kratsch regarding $ir_1$. Furthermore, we shall show that $\gamma_k^c(G)\leq{3k+1\over2}\gamma_k^t(G)-2k$ for each connected graph $G$, thereby generalizing a result of Favaron and Kratsch regarding $k=1$.

The minimum eccentric distance sum of trees with given distance k-domination number

2020 ◽  
Vol 12 (04) ◽  
pp. 2050052 ◽  
Author(s):  
Lidan Pei ◽  
Xiangfeng Pan
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Number Formula ◽  
Simple Connected Graph ◽  
Eccentric Distance ◽  
Distance Formula

Let [Formula: see text] be a positive integer and [Formula: see text] be a simple connected graph. The eccentric distance sum of [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the maximum distance from [Formula: see text] to any other vertex and [Formula: see text] is the sum of all distances from [Formula: see text]. A set [Formula: see text] is a distance [Formula: see text]-dominating set of [Formula: see text] if for every vertex [Formula: see text], [Formula: see text] for some vertex [Formula: see text]. The minimum cardinality among all distance [Formula: see text]-dominating sets of [Formula: see text] is called the distance [Formula: see text]-domination number [Formula: see text] of [Formula: see text]. In this paper, the trees among all [Formula: see text]-vertex trees with distance [Formula: see text]-domination number [Formula: see text] having the minimal eccentric distance sum are determined.

On the minimum vertex covering transversal dominating sets in graphs and their classification

2017 ◽  
Vol 09 (05) ◽  
pp. 1750069 ◽  
Author(s):  
R. Vasanthi ◽  
K. Subramanian
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Vertex Covering ◽  
Number Formula ◽  
Covering Set ◽  
Simple Connected Graph

Let [Formula: see text] be a simple and connected graph. A dominating set [Formula: see text] is said to be a vertex covering transversal dominating set if it intersects every minimum vertex covering set of [Formula: see text]. The vertex covering transversal domination number [Formula: see text] is the minimum cardinality among all vertex covering transversal dominating sets of [Formula: see text]. A vertex covering transversal dominating set of minimum cardinality [Formula: see text] is called a minimum vertex covering transversal dominating set or simply a [Formula: see text]-set. In this paper, we prove some general theorems on the vertex covering transversal domination number of a simple connected graph. We also provide some results about [Formula: see text]-sets and try to classify those sets based on their intersection with the minimum vertex covering sets.

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 $H$.

scholarly journals The edge-to-edge geodetic domination number of a graph

2021 ◽  
Vol 40 (3) ◽  
pp. 635-658
Author(s):  
J. John ◽  
V. Sujin Flower
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Upper Bounds ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
General Properties ◽  
Geodetic Number ◽  
Geodetic Set ◽  
Positive Integers

Let G = (V, E) be a connected graph with at least three vertices. A set S ⊆ E(G) is called an edge-to-edge geodetic dominating set of G if S is both an edge-to-edge geodetic set of G and an edge dominating set of G. The edge-to-edge geodetic domination number γgee(G) of G is the minimum cardinality of its edge-to-edge geodetic dominating sets. Some general properties satisfied by this concept are studied. Connected graphs of size m with edge-to-edge geodetic domination number 2 or m or m − 1 are characterized. We proved that if G is a connected graph of size m ≥ 4 and Ḡ is also connected, then 4 ≤ γgee(G) + γgee(Ḡ) ≤ 2m − 2. Moreover we characterized graphs for which the lower and the upper bounds are sharp. It is shown that, for every pair of positive integers a, b with 2 ≤ a ≤ b, there exists a connected graph G with gee(G) = a and γgee(G) = b. Also it is shown that, for every pair of positive integers a and b with 2 < a ≤ b, there exists a connected graph G with γe(G) = a and γgee(G) = b, where γe(G) is the edge domination number of G and gee(G) is the edge-to-edge geodetic number of G.

scholarly journals Adaptive Identification of Sets of Vertices in Graphs

2012 ◽  
Vol Vol. 14 no. 1 (Combinatorics) ◽  
Author(s):  
Ville Junnila
Keyword(s):  
Sensor Networks ◽  
Fault Detection ◽  
Adaptive Algorithm ◽  
Adaptive Algorithms ◽  
Multiprocessor Systems ◽  
Identifying Codes ◽  
Adaptive Identification ◽  
International Audience

Combinatorics International audience In this paper, we consider a concept of adaptive identification of vertices and sets of vertices in different graphs, which was recently introduced by Ben-Haim, Gravier, Lobstein and Moncel (2008). The motivation for adaptive identification comes from applications such as sensor networks and fault detection in multiprocessor systems. We present an optimal adaptive algorithm for identifying vertices in cycles. We also give efficient adaptive algorithms for identifying sets of vertices in different graphs such as cycles, king lattices and square lattices. Adaptive identification is also considered in Hamming spaces, which is one of the most widely studied graphs in the field of identifying codes.

Some results for the two disjoint connected dominating sets problem

2019 ◽  
Vol 11 (06) ◽  
pp. 1950065
Author(s):  
Xianliang Liu ◽  
Zishen Yang ◽  
Wei Wang
Keyword(s):  
Upper Bound ◽  
Connected Graph ◽  
Dominating Set ◽  
Sufficient Condition ◽  
Dominating Sets ◽  
Cubic Graph ◽  
Connected Dominating Sets ◽  
Minimum Cardinality ◽  
Vertex Set ◽  
Dominating Set Problem

As a variant of minimum connected dominating set problem, two disjoint connected dominating sets (DCDS) problem is to ask whether there are two DCDS [Formula: see text] in a connected graph [Formula: see text] with [Formula: see text] and [Formula: see text], and if not, how to add an edge subset with minimum cardinality such that the new graph has a pair of DCDS. The two DCDS problem is so hard that it is NP-hard on trees. In this paper, if the vertex set [Formula: see text] of a connected graph [Formula: see text] can be partitioned into two DCDS of [Formula: see text], then it is called a DCDS graph. First, a necessary but not sufficient condition is proposed for cubic (3-regular) graph to be a DCDS graph. To be exact, if a cubic graph is a DCDS graph, there are at most four disjoint triangles in it. Next, if a connected graph [Formula: see text] is a DCDS graph, a simple but nontrivial upper bound [Formula: see text] of the girth [Formula: see text] is presented.

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 Cost Effective Domination in the Join, Corona and Composition of Graphs

2019 ◽  
Vol 12 (3) ◽  
pp. 978-998
Author(s):  
Ferdinand P. Jamil ◽  
Hearty Nuenay Maglanque
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Cost Effective ◽  
Minimal Cost ◽  
Dominating Sets ◽  
Maximum Cardinality ◽  
Minimum Cardinality ◽  
The Cost ◽  
Domination Numbers

Let $G$ be a connected graph. A cost effective dominating set in a graph $G$ is any set $S$ of vertices in $G$ satisfying the condition that each vertex in $S$ is adjacent to at least as many vertices outside $S$ as inside $S$ and every vertex outside $S$ is adjacent to at least one vertex in $S$. The minimum cardinality of a cost effective dominating set is the cost effective domination number of $G$. The maximum cardinality of a cost effective dominating set is the upper cost effective domination number of $G$. A cost effective dominating set is said to be minimal if it does not contain a proper subset which is itself a cost effective dominating in $G$. The maximum cardinality of a minimal cost effective dominating set in a graph $G$ is the minimal cost effective domination number of $G$.In this paper, we characterized the cost effective dominating sets in the join, corona and composition of graphs. As direct consequences, we the bounds or the exact cost effective domination numbers, minimal cost effective domination numbers and upper cost effective domination numbers of these graphs were obtained.

Sign in / Sign up

Export Citation Format

Share Document