scholarly journals On Eternal Domination of Generalized J s , m

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Ramy Shaheen ◽  
Mohammad Assaad ◽  
Ali Kassem
Keyword(s):  
Infinite Series ◽  
Dominating Set ◽  
Domination Number ◽  
Adjacent Vertex ◽  
Dominating Set Problem ◽  
Jahangir Graph ◽  
Domination Numbers

An eternal dominating set of a graph G is a set of guards distributed on the vertices of a dominating set so that each vertex can be occupied by one guard only. These guards can defend any infinite series of attacks, an attack is defended by moving one guard along an edge from its position to the attacked vertex. We consider the “all guards move” of the eternal dominating set problem, in which one guard has to move to the attacked vertex, and all the remaining guards are allowed to move to an adjacent vertex or stay in their current positions after each attack in order to form a dominating set on the graph and at each step can be moved after each attack. The “all guards move model” is called the m -eternal domination model. The size of the smallest m -eternal dominating set is called the m -eternal domination number and is denoted by γ m ∞ G . In this paper, we find the domination number of Jahangir graph J s , m for s ≡ 1 , 2   mod   3 , and the m -eternal domination numbers of J s , m for s , m are arbitraries.

scholarly journals Eternal Domination of Generalized Petersen Graph

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Ramy Shaheen ◽  
Ali Kassem
Keyword(s):  
Infinite Series ◽  
Dominating Set ◽  
Domination Number ◽  
Upper Bounds ◽  
Adjacent Vertex ◽  
Petersen Graph ◽  
Generalized Petersen Graph ◽  
Dominating Set Problem

An eternal dominating set of a graph G is a set of guards distributed on the vertices of a dominating set so that each vertex can be occupied by one guard only. These guards can defend any infinite series of attacks; an attack is defended by moving one guard along an edge from its position to the attacked vertex. We consider the “all guards move” of the eternal dominating set problem, in which one guard has to move to the attacked vertex and all the remaining guards are allowed to move to an adjacent vertex or stay in their current positions after each attack in order to form a dominating set on the graph and at each step can be moved after each attack. The “all guards move model” is called the m -eternal domination model. The size of the smallest m -eternal dominating set is called the m -eternal domination number and is denoted by γ m ∞ G . In this paper, we find γ m ∞ P n , 1 and γ m ∞ P n , 3 for n ≡ 0   mod   4 . We also find upper bounds for γ m ∞ P n , 2 and γ m ∞ P n , 3 when n is arbitrary.

scholarly journals Fault Tolerance and 2-Domination in Certain Interconnection Networks

2019 ◽  
Vol 11 (2) ◽  
pp. 181
Author(s):  
Lian Chen ◽  
Xiujun Zhang
Keyword(s):  
Fault Tolerance ◽  
Sensor Network ◽  
Interconnection Networks ◽  
Dominating Set ◽  
Minimum Weight ◽  
Domination Number ◽  
Dominating Set Problem ◽  
Dominating Function ◽  
Domination Numbers

A graph could be understood as a sensor network, in which the vertices represent the sensors and two vertices are adjacent if and only if the corresponding devices can communicate with each other. For a network G, a 2-dominating function on G is a function f : V(G) → [0, 1] such that each vertex assigned with 0 has at least two neighbors assigned with 1. The weight of f is Σ_u∈V(G) f (u), and the minimum weight over all 2-dominating functions is the 2-domination number of G. The 2-dominating set problem consists of finding the 2-domination number of a graph and it was proposed to model the fault tolerance of a sensor network. In this paper, we determined substantial 2-domination numbers of 2-dimensional meshes, cylinders, tori and hypercubes.

scholarly journals Hop Domination in Graphs-II

2015 ◽  
Vol 23 (2) ◽  
pp. 187-199
Author(s):  
C. Natarajan ◽  
S.K. Ayyaswamy
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Total Domination ◽  
Minimum Cardinality ◽  
Unicyclic Graphs ◽  
The Family ◽  
Independent Domination ◽  
Domination In Graphs ◽  
Domination Numbers

Abstract Let G = (V;E) be a graph. A set S ⊂ V (G) is a hop dominating set of G if for every v ∈ V - S, there exists u ∈ S such that d(u; v) = 2. The minimum cardinality of a hop dominating set of G is called a hop domination number of G and is denoted by γh(G). In this paper we characterize the family of trees and unicyclic graphs for which γh(G) = γt(G) and γh(G) = γc(G) where γt(G) and γc(G) are the total domination and connected domination numbers of G respectively. We then present the strong equality of hop domination and hop independent domination numbers for trees. Hop domination numbers of shadow graph and mycielskian graph of graph are also discussed.

Weak and strong domination in thorn graphs

2019 ◽  
Vol 13 (04) ◽  
pp. 2050071
Author(s):  
Derya Doğan Durgun ◽  
Berna Lökçü
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Number Formula ◽  
Thorn Graphs ◽  
Domination Numbers

Let [Formula: see text] be a graph and [Formula: see text] A dominating set [Formula: see text] is a set of vertices such that each vertex of [Formula: see text] is either in [Formula: see text] or has at least one neighbor in [Formula: see text]. The minimum cardinality of such a set is called the domination number of [Formula: see text], [Formula: see text] [Formula: see text] strongly dominates [Formula: see text] and [Formula: see text] weakly dominates [Formula: see text] if (i) [Formula: see text] and (ii) [Formula: see text] A set [Formula: see text] is a strong-dominating set, shortly sd-set, (weak-dominating set, shortly wd-set) of [Formula: see text] if every vertex in [Formula: see text] is strongly (weakly) dominated by at least one vertex in [Formula: see text]. The strong (weak) domination number [Formula: see text] of [Formula: see text] is the minimum cardinality of an sd-set (wd-set). In this paper, we present weak and strong domination numbers of thorn graphs.

scholarly journals Graphs with Constant Sum of Domination and Inverse Domination Numbers

2012 ◽  
Vol 2012 ◽  
pp. 1-7 ◽  
Author(s):  
T. Tamizh Chelvam ◽  
T. Asir
Keyword(s):  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Complete Characterization ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Connected Graphs ◽  
Vertex Set ◽  
Domination Numbers

A subset D of the vertex set of a graph G, is a dominating set if every vertex in V−D is adjacent to at least one vertex in D. The domination number γ(G) is the minimum cardinality of a dominating set of G. A subset of V−D, which is also a dominating set of G is called an inverse dominating set of G with respect to D. The inverse domination number γ′(G) is the minimum cardinality of the inverse dominating sets. Domke et al. (2004) characterized connected graphs G with γ(G)+γ′(G)=n, where n is the number of vertices in G. It is the purpose of this paper to give a complete characterization of graphs G with minimum degree at least two and γ(G)+γ′(G)=n−1.

scholarly journals Exponential Domination in Subcubic Graphs

10.37236/5711 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Stéphane Bessy ◽  
Pascal Ochem ◽  
Dieter Rautenbach
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Discrete Math ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
Minimum Order ◽  
Natural Variant ◽  
Dominating Set Problem ◽  
Subcubic Graphs ◽  
Domination In Graphs

As a natural variant of domination in graphs, Dankelmann et al. [Domination with exponential decay, Discrete Math. 309 (2009) 5877-5883] introduced exponential domination, where vertices are considered to have some dominating power that decreases exponentially with the distance, and the dominated vertices have to accumulate a sufficient amount of this power emanating from the dominating vertices. More precisely, if $S$ is a set of vertices of a graph $G$, then $S$ is an exponential dominating set of $G$ if $\sum\limits_{v\in S}\left(\frac{1}{2}\right)^{{\rm dist}_{(G,S)}(u,v)-1}\geq 1$ for every vertex $u$ in $V(G)\setminus S$, where ${\rm dist}_{(G,S)}(u,v)$ is the distance between $u\in V(G)\setminus S$ and $v\in S$ in the graph $G-(S\setminus \{ v\})$. The exponential domination number $\gamma_e(G)$ of $G$ is the minimum order of an exponential dominating set of $G$.In the present paper we study exponential domination in subcubic graphs. Our results are as follows: If $G$ is a connected subcubic graph of order $n(G)$, then $$\frac{n(G)}{6\log_2(n(G)+2)+4}\leq \gamma_e(G)\leq \frac{1}{3}(n(G)+2).$$ For every $\epsilon>0$, there is some $g$ such that $\gamma_e(G)\leq \epsilon n(G)$ for every cubic graph $G$ of girth at least $g$. For every $0<\alpha<\frac{2}{3\ln(2)}$, there are infinitely many cubic graphs $G$ with $\gamma_e(G)\leq \frac{3n(G)}{\ln(n(G))^{\alpha}}$. If $T$ is a subcubic tree, then $\gamma_e(T)\geq \frac{1}{6}(n(T)+2).$ For a given subcubic tree, $\gamma_e(T)$ can be determined in polynomial time. The minimum exponential dominating set problem is APX-hard for subcubic graphs.

On Double Domination Numbers of Generalized Petersen Graphs

2016 ◽  
Vol 13 (10) ◽  
pp. 6514-6518
Author(s):  
Minhong Sun ◽  
Zehui Shao
Keyword(s):  
Programming Model ◽  
Dominating Set ◽  
Domination Number ◽  
Upper Bounds ◽  
Minimum Size ◽  
Generalized Petersen Graphs ◽  
Small Parameters ◽  
Petersen Graphs ◽  
Double Domination ◽  
Domination Numbers

A (total) double dominating set in a graph G is a subset S ⊆ V(G) such that each vertex in V(G) is (total) dominated by at least 2 vertices in S. The (total) double domination number of G is the minimum size of a (total) double dominating set of G. We determine the total double domination numbers and give upper bounds for double domination numbers of generalized Petersen graphs. By applying an integer programming model for double domination numbers of a graph, we have determined some exact values of double domination numbers of these generalized Petersen graphs with small parameters. The result shows that the given upper bounds match these exact values.

ON THE TWIN DOMINATION NUMBER IN GENERALIZED DE BRUIJN AND GENERALIZED KAUTZ DIGRAPHS

2010 ◽  
Vol 02 (02) ◽  
pp. 199-205 ◽  
Author(s):  
JYHMIN KUO
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Kautz Digraphs ◽  
De Bruijn ◽  
Kautz Digraph ◽  
De Bruijn Digraph ◽  
Domination Numbers

Let V and A denote the vertex and edge sets of a digraph G. A set T ⊆V is a twin dominating set of G if for every vertex v ∈ V - T, there exist u, w ∈ T (possibly u = w) such that arcs (u, v), (v, w) ∈ A. The twin domination number γ*(G) of G is the cardinality of a minimum twin dominating set of G. In this note, we investigate the twin domination numbers of generalized de Bruijn digraph and generalized Kautz digraph. The bounds of twin domination number of special generalized Kautz digraphs are given.

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.

scholarly journals Equitable Power Domination Number of Mycielskian of Certain Graphs

2018 ◽  
Vol 7 (4.10) ◽  
pp. 842 ◽  
Author(s):  
S. Banu Priya ◽  
A. Parthiban ◽  
N. Srinivasan
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Simple Graph ◽  
Adjacent Vertex ◽  
Minimum Cardinality ◽  
Power Domination ◽  
Vertex Set ◽  
The Difference

Let  be a simple graph with vertex set  and edge set . A set  is called a power dominating set (PDS), if every vertex   is observed by some vertices in  by using the following rules: (i) if a vertex  in  is in PDS, then it dominates itself and all the adjacent vertices of  and (ii) if an observed vertex  in   has  adjacent vertices and if   of these vertices are already observed, then the remaining one non-observed vertex is also observed by  in . A power dominating set    in   is said to be an equitable power dominating set (EPDS), if for every  there exists an adjacent vertex   such that the difference between the degree of  and degree of  is less than or equal to 1, i.e., . The minimum cardinality of an equitable power dominating set of  is called the equitable power domination number of  and denoted by . The Mycielskian of a graph  is the graph  with vertex set  where , and edge set  In this paper we investigate the equitable power domination number of Mycielskian of certain graphs. 

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