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### On the r-domination number of a graph

Jerrold R. Griggs, Joan P. Hutchinson
<span title="">1992</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
Hutchinson, On the r-domination number of a graph, Discrete Mathematics 101 (1992) 65-72.  ...  For r > 0, let the r-domination number of a graph, d" be the size of a smallest set of vertices such that every vertex of the graph is within distance r of a vertex in that set.  ...  Albertson and A. M. Dean for many helpful graph theoretic discussions and S. Wagon for help with the graphics.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0012-365x(92)90591-3">doi:10.1016/0012-365x(92)90591-3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ibc34vppzzcjpk26aaevgmfd7y">fatcat:ibc34vppzzcjpk26aaevgmfd7y</a> </span>
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### On the distance domination number of bipartite graphs [article]

D. A. Mojdeh, S. R. Musawi, E. Nazari
<span title="2018-05-03">2018</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In this note we give upper bound on the k-distance domination number of a connected bipartite graph and improve some results have been given like Theorem 2.1 and 2,7 in [Tian and Xu, A note on distance  ...  The minimum cardinality among all k-distance dominating sets of G is called the k-distance domination number of G.  ...  A set S ⊆ V is a dominating set if every vertex in V is either in S or is adjacent to a vertex in S. The domination number γ(G) is the minimum cardinality of a dominating set of G.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1805.01280v1">arXiv:1805.01280v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/grlhbdsolbczhaqfrrpdxhpwrm">fatcat:grlhbdsolbczhaqfrrpdxhpwrm</a> </span>
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### On the super domination number of lexicographic product graphs [article]

M. Dettlaff, M. Lemańska, J. A. Rodríguez-Velázquez, R. Zuazua
<span title="2017-03-17">2017</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
As a consequence of the study, we show that the problem of finding the super domination number of a graph is NP-Hard.  ...  In this article we obtain closed formulas and tight bounds for the super dominating number of lexicographic product graphs in terms of invariants of the factor graphs involved in the product.  ...  Section 2 covers basic results on the super domination number of a graph, including a characterization of graphs of order n with γ sp (G) = n − 1.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1703.06034v1">arXiv:1703.06034v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/qcbbqvs5lzhapkg3w36f32uzey">fatcat:qcbbqvs5lzhapkg3w36f32uzey</a> </span>
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### On the Roman domination number of a graph

O. Favaron, H. Karami, R. Khoeilar, S.M. Sheikholeslami
<span title="">2009</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
connected graph G of order n ≥ 3, γ R (G) + γ (G) 2 ≤ n, where γ (G) is the domination number of G.  ...  A Roman dominating function of a graph G is a labeling f : V (G) −→ {0, 1, 2} such that every vertex with label 0 has a neighbor with label 2. The Roman domination number et al. [E.J. Cockayne, P.A.  ...  Acknowledgement The fourth author's research was supported by the Research Office of Azarbaijan University of Tarbiat Moallem.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.disc.2008.09.043">doi:10.1016/j.disc.2008.09.043</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/v2afrlmu5veixgoesxl2hah3rm">fatcat:v2afrlmu5veixgoesxl2hah3rm</a> </span>
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### Bounds on the k-domination number of a graph

Ermelinda DeLaViña, Wayne Goddard, Michael A. Henning, Ryan Pepper, Emil R. Vaughan
<span title="">2011</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/deqidnohqjdu7gsln5vdm6obre" style="color: black;">Applied Mathematics Letters</a> </i> &nbsp;
We prove two bounds on the k-domination number of a graph, inspired by two conjectures of the computer program Graffiti.pc.  ...  The k-domination number of a graph is the cardinality of a smallest set of vertices such that every vertex not in the set is adjacent to at least k vertices of the set.  ...  The work of the first and fourth authors was supported in part by the United States Department of Defense and resources of the Extreme Scale Systems Center at Oak Ridge National Laboratory.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.aml.2011.01.013">doi:10.1016/j.aml.2011.01.013</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/gkudp7yb75gvbim6yuesmvkjge">fatcat:gkudp7yb75gvbim6yuesmvkjge</a> </span>
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### New Bounds on the Triple Roman Domination Number of Graphs

M. Hajjari, H. Abdollahzadeh Ahangar, R. Khoeilar, Z. Shao, S. M. Sheikholeslami, Firdous A. Shah
<span title="2022-01-04">2022</span> <i title="Hindawi Limited"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/5etzjnhqc5b3ricneysxmg2p6e" style="color: black;">Journal of Mathematics</a> </i> &nbsp;
In this paper, we derive sharp upper and lower bounds on the sum γ 3 R G + γ 3 R G ¯ and product γ 3 R G γ 3 R G ¯ , where G ¯ is the complement of graph G .  ...  We also show that for each tree T of order n ≥ 2 , γ 3 R T ≤ 3 n + s T / 2 and γ 3 R T ≥ ⌈ 4 n T + 2 − ℓ T / 3 ⌉ , where s T and ℓ T are the number of support vertices and leaves of T .  ...  □Now, we provide a lower bound on the triple Roman domination number of a tree T in terms of its order and the number of its leaves and support vertices.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1155/2022/9992618">doi:10.1155/2022/9992618</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/lbeornvmtbgsrh6o6cchb5rphe">fatcat:lbeornvmtbgsrh6o6cchb5rphe</a> </span>
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### On equality in an upper bound for the restrained and total domination numbers of a graph

Peter Dankelmann, David Day, Johannes H. Hattingh, Michael A. Henning, Lisa R. Markus, Henda C. Swart
<span title="">2007</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
The restrained domination number of G, denoted by r (G), is the minimum cardinality of an RDS of G. A set S ⊆ V is a total dominating set (TDS) if every vertex in V is adjacent to a vertex in S.  ...  The total domination number of a graph G without isolated vertices, denoted by t (G), is the minimum cardinality of a TDS of G. Let and denote the minimum and maximum degrees, respectively, in G.  ...  Introduction For a graph G=(V , E), a set S is a dominating set if every vertex in V \S has a neighbor in S. The domination number (G) is the minimum cardinality of a dominating set of G.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.disc.2007.03.003">doi:10.1016/j.disc.2007.03.003</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/obcz7nwicjhstj3iyfgrc4uzwy">fatcat:obcz7nwicjhstj3iyfgrc4uzwy</a> </span>
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### On (t,r) broadcast domination of directed graphs [article]

Pamela E. Harris, Peter Hollander, Erik Insko
<span title="2021-05-24">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
A dominating set of a graph G is a set of vertices that contains at least one endpoint of every edge on the graph. The domination number of G is the order of a minimum dominating set of G.  ...  Our main result explores the interval of values obtained by considering the directed (t,r) broadcast domination numbers of all orientations of a graph G.  ...  Similar to the standard domination number of a graph, the distance-k domination number of a graph G is the minimal cardinality of a distance-k dominating set of G.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2105.11317v1">arXiv:2105.11317v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/2bqc4u7axzebvoy6d26btkhyhu">fatcat:2bqc4u7axzebvoy6d26btkhyhu</a> </span>
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### Bibliography on domination in graphs and some basic definitions of domination parameters

<span title="">1990</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
Laskar, On domination and independent domination numbers of a graph, Discrete Math. 23 (1978) 73-76. R.B. Allan and R.  ...  Vizing, A bound on the external stability number of a graph, Dokl. Akad. Nauk SSSR 164 (1965) 729-731. H.B. Walikar, On star-partition number of a graph, manuscript, June 1979. H.B.  ...
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### Bibliography on Domination in Graphs and Some Basic Definitions of Domination Parameters [chapter]

<span title="">1991</span> <i title="Elsevier"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/e6ft7vn5prcbflhmcmkbrfkjfa" style="color: black;">Annals of Discrete Mathematics</a> </i> &nbsp;
Laskar, On domination and independent domination numbers of a graph, Discrete Math. 23 (1978) 73-76. R.B. Allan and R.  ...  Vizing, A bound on the external stability number of a graph, Dokl. Akad. Nauk SSSR 164 (1965) 729-731. H.B. Walikar, On star-partition number of a graph, manuscript, June 1979. H.B.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0167-5060(08)71054-9">doi:10.1016/s0167-5060(08)71054-9</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ez3slvi5zbdy3iyvoez4ain4ey">fatcat:ez3slvi5zbdy3iyvoez4ain4ey</a> </span>
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### On the independence and domination numbers of replacement product graphs

Jay Cummings, Christine A. Kelley
<span title="2016-03-02">2016</span> <i title="Mathematical Sciences Publishers"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/p6e2u6r7vnaq5ady7zgkb3kilu" style="color: black;">Involve. A Journal of Mathematics</a> </i> &nbsp;
In particular, we present results on the independence number, the domination number, and the total domination number of these graphs.  ...  The replacement product is a noncommutative graph operation that has been widely applied in many areas. One of its advantages over other graph products is its ability to produce sparse graphs.  ...  Acknowledgements The results in this paper were part of Cummings' undergraduate honors thesis, "On invariants of replacement product graphs", that was conducted under the supervision of Kelley.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.2140/involve.2016.9.181">doi:10.2140/involve.2016.9.181</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/yoq4krxo25e3raizb5knk4glaa">fatcat:yoq4krxo25e3raizb5knk4glaa</a> </span>
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### The inverse strong non-split r-domination number of a graph

B.K Ameenal, R Selvakumar
<span title="2010-09-06">2010</span> <i title="African Journals Online (AJOL)"> International Journal of Engineering, Science and Technology </i> &nbsp;
In this paper, we define the notions of inverse strong non-split r-dominating set and inverse strong non-split r-domination number γ′ sns r(G) of a graph G.  ...  We characterize graphs for which γ sns r(G) + γ′ sns r(G) = n, where γ sns r(G) is the strong non-split r-domination number of G. We get many bounds on γ′ sns r(G).  ...  Thus in this paper, we defined the Inverse strong non-split r-dominating set and the Inverse strong nonsplit r-domination number of a graph.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4314/ijest.v2i1.59102">doi:10.4314/ijest.v2i1.59102</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/fq7562dafbd2df52j5r6sif4tq">fatcat:fq7562dafbd2df52j5r6sif4tq</a> </span>
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### On Dominator Chromatic Number Of Radial Graph Of Some Graphs

Kalaivani R, Vijayalakshmi D
<span title="2019-12-30">2019</span> <i title="Eleyon Publishers"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/dzs4hvkqjbg4belpxclu5wpc4m" style="color: black;">Kongunadu Research Journal</a> </i> &nbsp;
A dominator coloring is a coloring of the vertices of a graph such that every vertex is either alone in its color class or adjacent to all vertices of at least one other class.  ...  In this paper, we obtain the Dominator Chromatic number of the Radial graph for the Central graph of Star graph, Super-radial graph for Middle graph of Cycle and Central graph of Path.  ...  For any n ≥ 2, the dominator chromatic number of radial graph of C(K1,n) = n + 1 i.e., 1 )) ( ( , 1   n K C R n d  Proof.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.26524/krj293">doi:10.26524/krj293</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/qnifr5e3qrggvnbrklhb3je35y">fatcat:qnifr5e3qrggvnbrklhb3je35y</a> </span>
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### On k-tuple and k-tuple total domination numbers of regular graphs [article]

Sharareh Alipour, Amir Jafari, Morteza Saghafian
<span title="2018-01-20">2018</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Also, we give an upper bound for the r-tuple dominating number of r-regular graphs.  ...  In this paper, we give a simple approach to compute an upper bound for (r-1)-tuple total domination number of r-regular graphs.  ...  Now we show that the (r − 1)-tuple dominating number of the incidence graph of a projective plane of order r − 1 is exactly 2r(r − 1).  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1709.01245v2">arXiv:1709.01245v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/k645vqca2bb2pg6tyoqmh3ttfe">fatcat:k645vqca2bb2pg6tyoqmh3ttfe</a> </span>
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### Restrained domination in signed graphs

Anisha Jean Mathias, V. Sangeetha, Mukti Acharya
<span title="2020-07-01">2020</span> <i title="Walter de Gruyter GmbH"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/msuamokr2nexlcxobb7k55f5fe" style="color: black;">Acta Universitatis Sapientiae: Mathematica</a> </i> &nbsp;
The main aim of this paper is to initiate a study on restrained domination in the realm of different classes of signed graphs.  ...  The set D having least cardinality is called minimum restrained dominating set and its cardinality is the restrained domination number of Σ denoted by γr(Σ).  ...  The restrained domination number of graph G denoted by γ r (G) is the smallest cardinality of a restrained dominating set of G.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.2478/ausm-2020-0010">doi:10.2478/ausm-2020-0010</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/2imr2ef6q5fzlbfavwwvxynida">fatcat:2imr2ef6q5fzlbfavwwvxynida</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200818131813/https://content.sciendo.com/downloadpdf/journals/ausm/12/1/article-p155.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/27/c2/27c243659e93acf02a40087125a1eb8f50d95346.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.2478/ausm-2020-0010"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>
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