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Journal articles on the topic 'Domination problem'

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Published: 4 June 2021

Last updated: 18 February 2022

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1

Sandven, Hallvard. "Systemic domination, social institutions and the coalition problem." Politics, Philosophy & Economics 19, no. 4 (2020): 382–402. http://dx.doi.org/10.1177/1470594x20927927.

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This article argues for a systemic conception of freedom as non-domination. It does so by engaging with the debate on the so-called coalition problem. The coalition problem arises because non-domination holds that groups can be agents of (dominating) power, while also insisting that freedom be robust. Consequently, it seems to entail that everyone is in a constant state of domination at the hands of potential groups. However, the problem can be dissolved by rejecting a ‘strict possibility’ standard for interpreting non-domination’s robustness requirement. Frank Lovett and Philip Pettit propose

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2

Kao, Mong-Jen, Chung-Shou Liao, and D. T. Lee. "Capacitated Domination Problem." Algorithmica 60, no. 2 (2009): 274–300. http://dx.doi.org/10.1007/s00453-009-9336-x.

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3

Klostermeyer, William F., Margaret Ellen Messinger, and Anders Yeo. "Dominating vertex covers: the vertex-edge domination problem." Discussiones Mathematicae Graph Theory 41, no. 1 (2021): 123. http://dx.doi.org/10.7151/dmgt.2175.

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4

Vaidya, S. K., and R. M. Pandit. "Some New Perspectives on Global Domination in Graphs." ISRN Combinatorics 2013 (August 20, 2013): 1–4. http://dx.doi.org/10.1155/2013/201654.

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A dominating set is called a global dominating set if it is a dominating set of a graph G and its complement G¯. Here we explore the possibility to relate the domination number of graph G and the global domination number of the larger graph obtained from G by means of various graph operations. In this paper we consider the following problem: Does the global domination number remain invariant under any graph operations? We present an affirmative answer to this problem and establish several results.

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5

Chang, Gerard J., and Shiow-Fen Hwang. "The edge domination problem." Discussiones Mathematicae Graph Theory 15, no. 1 (1995): 51. http://dx.doi.org/10.7151/dmgt.1006.

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6

Klostermeyer, William F. "A Dynamic Domination Problem." Journal of Discrete Mathematical Sciences and Cryptography 18, no. 6 (2015): 837–48. http://dx.doi.org/10.1080/09720529.2015.1026457.

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7

Panda, B. S., and D. Pradhan. "A linear time algorithm to compute a minimum restrained dominating set in proper interval graphs." Discrete Mathematics, Algorithms and Applications 07, no. 02 (2015): 1550020. http://dx.doi.org/10.1142/s1793830915500202.

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A set D ⊆ V is a restrained dominating set of a graph G = (V, E) if every vertex in V\D is adjacent to a vertex in D and a vertex in V\D. Given a graph G and a positive integer k, the restrained domination problem is to check whether G has a restrained dominating set of size at most k. The restrained domination problem is known to be NP-complete even for chordal graphs. In this paper, we propose a linear time algorithm to compute a minimum restrained dominating set of a proper interval graph. We present a polynomial time reduction that proves the NP-completeness of the restrained domination pr

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8

Meybodi, M. Alambardar, M. R. Hooshmandasl, P. Sharifani, and A. Shakiba. "Domination cover number of graphs." Discrete Mathematics, Algorithms and Applications 11, no. 02 (2019): 1950020. http://dx.doi.org/10.1142/s1793830919500204.

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A set [Formula: see text] for the graph [Formula: see text] is called a dominating set if any vertex [Formula: see text] has at least one neighbor in [Formula: see text]. Fomin et al. [Combinatorial bounds via measure and conquer: Bounding minimal dominating sets and applications, ACM Transactions on Algorithms (TALG) 5(1) (2008) 9] gave an algorithm for enumerating all minimal dominating sets with [Formula: see text] vertices in [Formula: see text] time. It is known that the number of minimal dominating sets for interval graphs and trees on [Formula: see text] vertices is at most [Formula: se

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9

Das, Sayani, and Sounaka Mishra. "Lower bounds on approximating some variations of vertex coloring problem over restricted graph classes." Discrete Mathematics, Algorithms and Applications 12, no. 06 (2020): 2050086. http://dx.doi.org/10.1142/s179383092050086x.

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A [Formula: see text] vertex coloring of a graph [Formula: see text] partitions the vertex set into [Formula: see text] color classes (or independent sets). In minimum vertex coloring problem, the aim is to minimize the number of colors used in a given graph. Here, we consider three variations of vertex coloring problem in which (i) each vertex in [Formula: see text] dominates a color class, (ii) each color class is dominated by a vertex and (iii) each vertex is dominating a color class and each color class is dominated by a vertex. These minimization problems are known as Min-Dominator-Colori

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10

Qiang, Xiaoli, Maryam Akhoundi, Zheng Kou, Xinyue Liu, and Saeed Kosari. "Novel Concepts of Domination in Vague Graphs with Application in Medicine." Mathematical Problems in Engineering 2021 (June 24, 2021): 1–10. http://dx.doi.org/10.1155/2021/6121454.

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VG can manage the uncertainty relevant to the inconsistent and indeterminate information of all real-world problems, in which FGs possibly will not succeed in bringing about satisfactory results. The previous definitions’ restrictions in FGs have made us present new definitions in VGs. A wide range of applications have been attributed to the domination in graph theory for several fields such as facility location problem, school bus routing, modeling biological networks, and coding theory. Therefore, in this research, we study several concepts of domination, such as restrained dominating set (R

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11

PRADHAN, D. "COMPLEXITY OF CERTAIN FUNCTIONAL VARIANTS OF TOTAL DOMINATION IN CHORDAL BIPARTITE GRAPHS." Discrete Mathematics, Algorithms and Applications 04, no. 03 (2012): 1250045. http://dx.doi.org/10.1142/s1793830912500450.

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In this paper, we consider minimum total domination problem along with two of its variations namely, minimum signed total domination problem and minimum minus total domination problem for chordal bipartite graphs. In the minimum total domination problem, the objective is to find a smallest size subset TD ⊆ V of a given graph G = (V, E) such that |TD∩NG(v)| ≥ 1 for every v ∈ V. In the minimum signed (minus) total domination problem for a graph G = (V, E), it is required to find a function f : V → {-1, 1} ({-1, 0, 1}) such that f(NG(v)) = ∑u∈NG(v)f(u) ≥ 1 for each v ∈ V, and the cost f(V) = ∑v∈V

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12

LEE, CHUAN-MIN, and CHENG-CHIEN LO. "On the Complexity of Reverse Minus and Signed Domination on Graphs." Journal of Interconnection Networks 15, no. 01n02 (2015): 1550008. http://dx.doi.org/10.1142/s0219265915500085.

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Motivated by the concept of reverse signed domination, we introduce the reverse minus domination problem on graphs, and study the reverse minus and signed domination problems from the algorithmic point of view. In this paper, we show that both the reverse minus and signed domination problems are polynomial-time solvable for strongly chordal graphs and distance-hereditary graphs, and are linear-time solvable for trees. For chordal graphs and bipartite planar graphs, however, we show that the decision problem corresponding to the reverse minus domination problem is NP-complete. For doubly chorda

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13

Saifudin, Ilham. "Power Domination Number On Shackle Operation with Points as Lingkage." JTAM | Jurnal Teori dan Aplikasi Matematika 4, no. 1 (2020): 1. http://dx.doi.org/10.31764/jtam.v4i1.1579.

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The Power dominating set is a minimum point of determination in a graph that can dominate the connected dots around it, with a minimum domination point. The smallest cardinality of a power dominating set is called a power domination number with the notation . The purpose of this study is to determine the Shackle operations graph value from several special graphs with a point as a link. The result operation graphs are: Shackle operation graph from Path graph , Shackle operation graph from Sikel graph , Shackle operation graph from Star graph . The method used in this paper is axiomatic deductiv

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14

FRAISTAT, SHAWN. "Domination and Care in Rousseau'sEmile." American Political Science Review 110, no. 4 (2016): 889–900. http://dx.doi.org/10.1017/s0003055416000472.

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Domination, understood as the abusive or capricious employment of power over others for the sake of one's own ends, is among the gravest threats to human freedom. Solving the problem of domination is a crucial normative challenge, and this article identifies in the work of Jean-Jacques Rousseau a promising and overlooked avenue for addressing it. I propose an interpretation of Rousseau'sEmilein which preventing domination requires moral education in the practice and value of care. This interpretation gives Rousseau new relevance as a theorist of domination. In connecting non-dominative to educ

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15

Kumar, J. Pavan, and P. Venkata Subba Reddy. "Algorithmic Aspects of Some Variants of Domination in Graphs." Analele Universitatii "Ovidius" Constanta - Seria Matematica 28, no. 3 (2020): 153–70. http://dx.doi.org/10.2478/auom-2020-0039.

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Abstract A set S ⊆ V is a dominating set in G if for every u ∈ V \ S, there exists v ∈ S such that (u, v) ∈ E, i.e., N[S] = V . A dominating set S is an isolate dominating set (IDS) if the induced subgraph G[S] has at least one isolated vertex. It is known that Isolate Domination Decision problem (IDOM) is NP-complete for bipartite graphs. In this paper, we extend this by showing that the IDOM is NP-complete for split graphs and perfect elimination bipartite graphs, a subclass of bipartite graphs. A set S ⊆ V is an independent set if G[S] has no edge. A set S ⊆ V is a secure dominating set of

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16

Kusraev, A. G. "Domination problem in Banach lattices." Mathematical Notes 100, no. 1-2 (2016): 66–79. http://dx.doi.org/10.1134/s0001434616070063.

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17

Hwang, Shiow-Fen, and Gerard J. Chang. "The k-neighbor domination problem." European Journal of Operational Research 52, no. 3 (1991): 373–77. http://dx.doi.org/10.1016/0377-2217(91)90172-r.

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18

Grinstead, Charles M., Bruce Hahne, and David Van Stone. "On the queen domination problem." Discrete Mathematics 86, no. 1-3 (1990): 21–26. http://dx.doi.org/10.1016/0012-365x(90)90345-i.

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19

Baklouti, Hamadi, and Mohamed Hajji. "Schur operators and domination problem." Positivity 21, no. 1 (2016): 35–48. http://dx.doi.org/10.1007/s11117-016-0400-x.

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20

Yen, Chain-Chin, and R. C. T. Lee. "The weighted perfect domination problem." Information Processing Letters 35, no. 6 (1990): 295–99. http://dx.doi.org/10.1016/0020-0190(90)90031-r.

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21

Chen, Xue-gang, and Shinya Fujita. "Downhill domination problem in graphs." Information Processing Letters 115, no. 6-8 (2015): 580–81. http://dx.doi.org/10.1016/j.ipl.2015.02.003.

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22

Bakhshesh, Davood. "Characterization of some classes of graphs with equal domination number and isolate domination number." Discrete Mathematics, Algorithms and Applications 12, no. 05 (2020): 2050065. http://dx.doi.org/10.1142/s1793830920500652.

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Let [Formula: see text] be a simple and undirected graph with vertex set [Formula: see text]. A set [Formula: see text] is called a dominating set of [Formula: see text], if every vertex in [Formula: see text] is adjacent to at least one vertex in [Formula: see text]. The minimum cardinality of a dominating set of [Formula: see text] is called the domination number of [Formula: see text], denoted by [Formula: see text]. A dominating set [Formula: see text] of [Formula: see text] is called isolate dominating, if the induced subgraph [Formula: see text] of [Formula: see text] contains at least o

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23

Dong, Wen, and Shi Qiao. "Several Strategies to Compute the Minimum Stochastic Dominating Sets in Graphs." Applied Mechanics and Materials 380-384 (August 2013): 1221–25. http://dx.doi.org/10.4028/www.scientific.net/amm.380-384.1221.

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On this paper, a simple introduction about the concept of the domination set in graphs and the research progress is presented. The definition of stochastic dominating set is introduced as well; We designed several strategies to compute the stochastic dominating set, and analyzed the domination rate using probability theory in the broadcasting model. These strategies have important theory value to solve network communication redundancy package problem.

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24

Kuo, Jyhmin, and Wei-Lun Wu. "Power domination in generalized undirected de Bruijn graphs and Kautz graphs." Discrete Mathematics, Algorithms and Applications 07, no. 01 (2015): 1550003. http://dx.doi.org/10.1142/s1793830915500032.

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To monitor an electric power system by placing as few phase measurement units (PMUs) as possible is closely related to the famous vertex cover problem and domination problem in graph theory. A set P is a power dominating set (PDS) of a graph G = (V, E), if every vertex and every edge in the system is observed following the observation rules of power system monitoring. The minimum cardinality of a PDS of a graph G is the power domination number γp(G). In this paper, we determine the upper bounds of power domination number of generalized undirected de Bruijn graphs and generalized undirected Kau

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25

Sharma, Amit, and P. Venkata Subba Reddy. "Algorithmic Aspects of Outer-Independent Total Roman Domination in Graphs." International Journal of Foundations of Computer Science 32, no. 03 (2021): 331–39. http://dx.doi.org/10.1142/s0129054121500180.

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For a simple, undirected graph [Formula: see text], a function [Formula: see text] which satisfies the following conditions is called an outer-independent total Roman dominating function (OITRDF) of [Formula: see text] with weight [Formula: see text]. (C1) For all [Formula: see text] with [Formula: see text] there exists a vertex [Formula: see text] such that [Formula: see text] and [Formula: see text], (C2) The induced subgraph with vertex set [Formula: see text] has no isolated vertices and (C3) The induced subgraph with vertex set [Formula: see text] is independent. For a graph [Formula: se

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26

Lafont, Jean-François, and Christoforos Neofytidis. "Steenrod problem and the domination relation." Topology and its Applications 255 (March 2019): 32–40. http://dx.doi.org/10.1016/j.topol.2018.12.016.

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27

Brešar, Boštjan, Paul Dorbec, Sandi Klavžar, Gašper Košmrlj, and Gabriel Renault. "Complexity of the game domination problem." Theoretical Computer Science 648 (October 2016): 1–7. http://dx.doi.org/10.1016/j.tcs.2016.07.025.

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28

Kinsley, A. Anto, and S. Somasundaram. "Domination based algorithm tok-center problem." Journal of Discrete Mathematical Sciences and Cryptography 9, no. 3 (2006): 403–16. http://dx.doi.org/10.1080/09720529.2006.10698087.

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29

Cockayne, E. J., and S. T. Hedetniemi. "On the diagonal queens domination problem." Journal of Combinatorial Theory, Series A 42, no. 1 (1986): 137–39. http://dx.doi.org/10.1016/0097-3165(86)90012-9.

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30

Baklouti, Hamadi, and Mohamed Hajji. "Disjointly improjective operators and domination problem." Indagationes Mathematicae 28, no. 6 (2017): 1175–82. http://dx.doi.org/10.1016/j.indag.2017.09.002.

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31

Chang, Chan-Wei, David Kuo, Sheng-Chyang Liaw, and Jing-Ho Yan. "$$F_{3}$$ -domination problem of graphs." Journal of Combinatorial Optimization 28, no. 2 (2012): 400–413. http://dx.doi.org/10.1007/s10878-012-9563-y.

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32

Jha, Anupriya, D. Pradhan, and S. Banerjee. "The secure domination problem in cographs." Information Processing Letters 145 (May 2019): 30–38. http://dx.doi.org/10.1016/j.ipl.2019.01.005.

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33

PANDA, B. S., and S. PAUL. "CONNECTED LIAR'S DOMINATION IN GRAPHS: COMPLEXITY AND ALGORITHMS." Discrete Mathematics, Algorithms and Applications 05, no. 04 (2013): 1350024. http://dx.doi.org/10.1142/s1793830913500249.

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A subset L ⊆ V of a graph G = (V, E) is called a connected liar's dominating set of G if (i) for all v ∈ V, |NG[v] ∩ L| ≥ 2, (ii) for every pair u, v ∈ V of distinct vertices, |(NG[u]∪NG[v])∩L| ≥ 3, and (iii) the induced subgraph of G on L is connected. In this paper, we initiate the algorithmic study of minimum connected liar's domination problem by showing that the corresponding decision version of the problem is NP-complete for general graph. Next we study this problem in subclasses of chordal graphs where we strengthen the NP-completeness of this problem for undirected path graph and prove

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34

Shaheen, Ramy, Mohammad Assaad, and Ali Kassem. "On Eternal Domination of Generalized J s , m." Journal of Applied Mathematics 2021 (March 17, 2021): 1–7. http://dx.doi.org/10.1155/2021/8882598.

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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

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35

Chen, Lian, and Xiujun Zhang. "Fault Tolerance and 2-Domination in Certain Interconnection Networks." Journal of Mathematics Research 11, no. 2 (2019): 181. http://dx.doi.org/10.5539/jmr.v11n2p181.

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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) &rarr; [0, 1] such that each vertex assigned with 0 has at least two neighbors assigned with 1. The weight of f is &Sigma;_u&isin;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 w

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36

Miao, Fang, Wenjie Fan, Mustapha Chellali, Rana Khoeilar, Seyed Mahmoud Sheikholeslami, and Marzieh Soroudi. "On Two Open Problems on Double Vertex-Edge Domination in Graphs." Mathematics 7, no. 11 (2019): 1010. http://dx.doi.org/10.3390/math7111010.

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A vertex v of a graph G = ( V , E ) , ve-dominates every edge incident to v, as well as every edge adjacent to these incident edges. A set S ⊆ V is a double vertex-edge dominating set if every edge of E is ve-dominated by at least two vertices of S. The double vertex-edge domination number γ d v e ( G ) is the minimum cardinality of a double vertex-edge dominating set in G. A subset S ⊆ V is a total dominating set (respectively, a 2-dominating set) if every vertex in V has a neighbor in S (respectively, every vertex in V - S has at least two neighbors in S). The total domination number γ t ( G

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37

Dobrinen, Natasha L., and Stephen G. Simpson. "Almost everywhere domination." Journal of Symbolic Logic 69, no. 3 (2004): 914–22. http://dx.doi.org/10.2178/jsl/1096901775.

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Abstract.A Turing degree a is said to be almost everywhere dominating if, for almost all X ∈ 2ω with respect to the “fair coin” probability measure on 2ω, and for all g: ω → ω Turing reducible to X, there exists f: ω → ω of Turing degree a which dominates g. We study the problem of characterizing the almost everywhere dominating Turing degrees and other, similarly defined classes of Turing degrees. We relate this problem to some questions in the reverse mathematics of measure theory.

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38

Cabrera-Martínez, Abel, Juan Carlos Hernández-Gómez, Ernesto Parra-Inza, and José María Sigarreta Almira. "On the Total Outer k-Independent Domination Number of Graphs." Mathematics 8, no. 2 (2020): 194. http://dx.doi.org/10.3390/math8020194.

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A set of vertices of a graph G is a total dominating set if every vertex of G is adjacent to at least one vertex in such a set. We say that a total dominating set D is a total outer k-independent dominating set of G if the maximum degree of the subgraph induced by the vertices that are not in D is less or equal to k − 1 . The minimum cardinality among all total outer k-independent dominating sets is the total outer k-independent domination number of G. In this article, we introduce this parameter and begin with the study of its combinatorial and computational properties. For instance, we give

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39

Amjadi, J., N. Dehgardi, N. Mohammadi, S. M. Sheikholeslami, and L. Volkmann. "Independent 2-rainbow domination in trees." Asian-European Journal of Mathematics 08, no. 02 (2015): 1550035. http://dx.doi.org/10.1142/s1793557115500357.

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A 2-rainbow dominating function (2RDF) on a graph G is a function f from the vertex set V(G) to the set of all subsets of the set {1, 2} such that for any vertex v ∈ V(G) with f(v) = ∅ the condition ⋃u∈N(v)f(u) = {1, 2} is fulfilled. A 2RDF f is independent 2-rainbow dominating function (I2RDF) if no two vertices assigned nonempty sets are adjacent. The weight of a 2RDF f is the value ω(f) = ∑v∈V |f(v)|. The 2-rainbow domination number γr2(G) (respectively, the independent 2-rainbow domination number ir2(G)) is the minimum weight of a 2RDF (respectively, I2RDF) on G. M. Chellali and N. Jafari

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40

Zhang, Xiujun, Zehui Shao, and Hong Yang. "The [a,b]-domination and [a,b]-total Domination of Graphs." Journal of Mathematics Research 9, no. 3 (2017): 38. http://dx.doi.org/10.5539/jmr.v9n3p38.

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A subset $S$ of the vertices of $G = (V, E)$ is an $[a, b]$-set if for every vertex $v$ not in $S$ we have the number of neighbors of $v$ in $S$ is between $a$ and $b$ for non-negative integers $a$ and $b$, that is, every vertex $v$ not in $S$ is adjacent to at least $a$ but not more than $b$ vertices in $S$. The minimum cardinality of an $[a, b]$-set of $G$ is called the $[a, b]$-domination number of $G$. The $[a, b]$-domination problem is to determine the $[a, b]$-domination number of a graph. In this paper, we show that the [2,b]-domination problem is NP-complete for $b$ at least $3$, and t

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41

Dorokhina, Daria M. "THE PROBLEM OF POLITICAL DOMINATION(S. FRANK)." RSUH/RGGU Bulletin. Series Philosophy. Social Studies. Art Studies, no. 2 (2019): 24–34. http://dx.doi.org/10.28995/2073-6401-2019-2-24-34.

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42

Lan, James K., and Gerard Jennhwa Chang. "On the mixed domination problem in graphs." Theoretical Computer Science 476 (March 2013): 84–93. http://dx.doi.org/10.1016/j.tcs.2012.11.035.

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43

Liu, Zeyu, Xueping Li, and Anahita Khojandi. "On the k-Strong Roman Domination Problem." Discrete Applied Mathematics 285 (October 2020): 227–41. http://dx.doi.org/10.1016/j.dam.2020.05.018.

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44

Pliev, Marat. "Domination problem for narrow orthogonally additive operators." Positivity 21, no. 1 (2016): 23–33. http://dx.doi.org/10.1007/s11117-016-0401-9.

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45

Kao, Mong-Jen, Han-Lin Chen, and D. T. Lee. "Capacitated Domination: Problem Complexity and Approximation Algorithms." Algorithmica 72, no. 1 (2013): 1–43. http://dx.doi.org/10.1007/s00453-013-9844-6.

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46

Lin, Ching-Chi, Keng-Chu Ku, and Chan-Hung Hsu. "Paired-Domination Problem on Distance-Hereditary Graphs." Algorithmica 82, no. 10 (2020): 2809–40. http://dx.doi.org/10.1007/s00453-020-00705-7.

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47

Su, Zishan, Chun Su, Zhishui Hu, and Jie Liu. "On domination problem of non-negative distributions." Frontiers of Mathematics in China 4, no. 4 (2009): 681–96. http://dx.doi.org/10.1007/s11464-009-0040-6.

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48

Cabrera Martínez, Abel, Luis P. Montejano, and Juan A. Rodríguez-Velázquez. "Total Weak Roman Domination in Graphs." Symmetry 11, no. 6 (2019): 831. http://dx.doi.org/10.3390/sym11060831.

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Given a graph G = ( V , E ) , a function f : V → { 0 , 1 , 2 , ⋯ } is said to be a total dominating function if ∑ u ∈ N ( v ) f ( u ) > 0 for every v ∈ V , where N ( v ) denotes the open neighbourhood of v. Let V i = { x ∈ V : f ( x ) = i } . We say that a function f : V → { 0 , 1 , 2 } is a total weak Roman dominating function if f is a total dominating function and for every vertex v ∈ V 0 there exists u ∈ N ( v ) ∩ ( V 1 ∪ V 2 ) such that the function f ′ , defined by f ′ ( v ) = 1 , f ′ ( u ) = f ( u ) - 1 and f ′ ( x ) = f ( x ) whenever x ∈ V ∖ { u , v } , is a total dominating functi

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Almerich-Chulia, Ana, Abel Cabrera Martínez, Frank Angel Hernández Mira, and Pedro Martin-Concepcion. "From Total Roman Domination in Lexicographic Product Graphs to Strongly Total Roman Domination in Graphs." Symmetry 13, no. 7 (2021): 1282. http://dx.doi.org/10.3390/sym13071282.

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Let G be a graph with no isolated vertex and let N(v) be the open neighbourhood of v∈V(G). Let f:V(G)→{0,1,2} be a function and Vi={v∈V(G):f(v)=i} for every i∈{0,1,2}. We say that f is a strongly total Roman dominating function on G if the subgraph induced by V1∪V2 has no isolated vertex and N(v)∩V2≠∅ for every v∈V(G)\V2. The strongly total Roman domination number of G, denoted by γtRs(G), is defined as the minimum weight ω(f)=∑x∈V(G)f(x) among all strongly total Roman dominating functions f on G. This paper is devoted to the study of the strongly total Roman domination number of a graph and i

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Shaheen, Ramy, and Ali Kassem. "Eternal Domination of Generalized Petersen Graph." Journal of Applied Mathematics 2021 (January 25, 2021): 1–10. http://dx.doi.org/10.1155/2021/6627272.

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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

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