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On the number of edges in graphs with a given weakly connected domination number

Laura A. Sanchis
<span title="">2002</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
We show in this paper that the maximum number of edges that a graph with n vertices and weakly connected domination number equal to d ¿ 3 can have is ( n−d+1 2 ).  ...  A dominating set is weakly connected if the edges not incident on any vertices of the dominating set do not separate the graph (Discrete Math. 167-168 (1997) 261).  ...  Acknowledgements The author is grateful to Jerrold Grossman for suggesting this problem.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0012-365x(01)00474-5">doi:10.1016/s0012-365x(01)00474-5</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/tbklmhvpgjh37lsp2ctpsnp5qu">fatcat:tbklmhvpgjh37lsp2ctpsnp5qu</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20171001174725/http://publisher-connector.core.ac.uk/resourcesync/data/elsevier/pdf/a7c/aHR0cDovL2FwaS5lbHNldmllci5jb20vY29udGVudC9hcnRpY2xlL3BpaS9zMDAxMjM2NXgwMTAwNDc0NQ%3D%3D.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/46/63/46631ef8a0f76e20125ddb8f1c0aac02ea1c7c46.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0012-365x(01)00474-5"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

Analysis of d-Hop Dominating Set Problem for Directed Graph with Indegree Bounded by One [article]

Joydeep Banerjee, Arun Das, Arunabha Sen
<span title="2017-01-25">2017</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In this short paper we perform computational complexity analysis of minimum d-hop dominating set problem for directed graphs with in-degree bounded by 1.  ...  The optimum solution of the problem can be found polynomially by exploiting certain properties of the graph under consideration.  ...  All the graphs have at least one node with in-degree greater than one (marked as white) Property 2.2 The number of directed edges of the graph G D is upper bounded by |V D | as all nodes have in-degree  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1404.6890v2">arXiv:1404.6890v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/cpuj2bbj6bfwjllxigcarldfsy">fatcat:cpuj2bbj6bfwjllxigcarldfsy</a> </span>
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Dominating Sets and Spanning Tree based Clustering Algorithms for Mobile Ad hoc Networks

R Krishnam, Suresh Varma
<span title="">2011</span> <i title="The Science and Information Organization"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2yzw5hsmlfa6bkafwsibbudu64" style="color: black;">International Journal of Advanced Computer Science and Applications</a> </i> &nbsp;
Dominating sets and Spanning Tree based MANET clustering algorithms were suitable in a medium size network with respect to time and message complexities.  ...  Clustering algorithms in mobile ad hoc networks builds a virtual backbone for network nodes. Dominating sets and Spanning tree are widely used in clustering networks.  ...  Weakly Connected Dominating Sets (WCDS) WCDS [10] [12] is Sw is a subset S of a graph G that contains the vertices of S, their neighbors and all edges of the original graph G with at least one endpoint  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.14569/ijacsa.2011.020213">doi:10.14569/ijacsa.2011.020213</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/73243rrzzfco3awdglldmdwdm4">fatcat:73243rrzzfco3awdglldmdwdm4</a> </span>
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Page 6602 of Mathematical Reviews Vol. , Issue 97K [page]

<span title="">1997</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
97k:05122 such as the domination number, the edge domination number, the total domination number, the connected domination number, the vertex-edge weak/strong domination number and the edge-vertex weak  ...  /strong domination number of graphs under the transfor- mation in which one edge or one pendant vertex is deleted are discussed.  ... 
<span class="external-identifiers"> </span>
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Randomised Algorithms for Finding Small Weakly-Connected Dominating Sets of Regular Graphs [chapter]

William Duckworth, Bernard Mans
<span title="">2003</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
A weakly-connected dominating set, W, of a graph, G, is a dominating set such that the subgraph consisting of V (G) and all edges incident with vertices in W is connected.  ...  A weakly-connected dominating set, W, of a graph, G, is a dominating set such that the subgraph consisting of V (G) and all edges incident with vertices in W is connected.  ...  [7] introduced the concept of a weakly-connected independent dominating set, i.e., a weakly-connected dominating set, I, of a graph, G, such that no two vertices in I are connected by an edge of G.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/3-540-44849-7_15">doi:10.1007/3-540-44849-7_15</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/632saeu6xve7tk6ud5fd6pwefq">fatcat:632saeu6xve7tk6ud5fd6pwefq</a> </span>
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Weakly connected independent domination in the join of graphs

Rene E. Leonida
<span title="">2013</span> <i title="Hikari, Ltd."> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/4p3snko5aze6pfbct5pvdfu7by" style="color: black;">International Mathematical Forum</a> </i> &nbsp;
In this paper we characterized the weakly connected independent dominating sets in the join of two graphs and as a consequence, the weakly connected independent domination number is obtained.  ...  A subset S of V (G) is called weakly connected if the subgraph S w = (N G [S], E W ) weakly induced by S, is connected, where E W is the set of all edges with at least one vertex in S. . . . . . . .  ...  The weakly connected domination number γ w (G) of G is the smallest cardinality of a weakly connected dominating set of G.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.12988/imf.2013.39170">doi:10.12988/imf.2013.39170</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ubptovgpybb7xnaamtrh37ksoe">fatcat:ubptovgpybb7xnaamtrh37ksoe</a> </span>
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Fast distributed algorithms for (weakly) connected dominating sets and linear-size skeletons

Devdatt Dubhashi, Alessandro Mei, Alessandro Panconesi, Jaikumar Radhakrishnan, Aravind Srinivasan
<span title="">2005</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/p6ovb2qpkfenhmb7mcksobrcxq" style="color: black;">Journal of computer and system sciences (Print)</a> </i> &nbsp;
Given a network, we first show how to compute connected and weakly connected dominating sets whose size is at most O(log ) times the optimum, being the maximum degree of the input network.  ...  (Given a dominating set S, a dominator of a vertex u is any v ∈ S such that the distance between u and v is at most one.) We also show our time bounds to be essentially optimal.  ...  Acknowledgments We thank Samir Khuller, the SODA 2003 referees, and the journal referees for their helpful comments.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.jcss.2005.04.002">doi:10.1016/j.jcss.2005.04.002</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/hnrofhi3fzaaxjpbzv2nqxzmo4">fatcat:hnrofhi3fzaaxjpbzv2nqxzmo4</a> </span>
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Page 722 of Mathematical Reviews Vol. , Issue 98B [page]

<span title="">1998</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
Summary: “A weakly connected dominating set for a connected graph is a dominating set D of vertices of the graph such that the edges not incident to any vertex in D do not separate the graph.  ...  (SA-RAND; Johannesbury); Hedetniemi, Stephen T. (1-CLEM-C; Clemson, SC); McRae, Alice A. (1-APLS-C; Boone, NC) On weakly connected domination in graphs.  ... 
<span class="external-identifiers"> </span>
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Approximating minimum size weakly-connected dominating sets for clustering mobile ad hoc networks

Yuanzhu Peter Chen, Arthur L. Liestman
<span title="">2002</span> <i title="ACM Press"> Proceedings of the 3rd ACM international symposium on Mobile ad hoc networking &amp; computing - MobiHoc &#39;02 </i> &nbsp;
We present a series of approximation algorithms for finding a small weakly-connected dominating set (WCDS) in a given graph to be used in clustering mobile ad hoc networks.  ...  Comparisons between our work and some previous work (CDS-based) are also given in terms of the size of resultant dominating sets and graph connectivity degradation.  ...  Figure 7 depicts the situation after 2 iterations on the given graph. In the figure, the five circled vertices are a minimum weakly-connected dominating set (OP T ).  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1145/513819.513821">doi:10.1145/513819.513821</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/wljhsxmlovaehpp2ly5tz7scvy">fatcat:wljhsxmlovaehpp2ly5tz7scvy</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170810111157/http://www.cs.mun.ca/~yzchen/papers/mobihoc02.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/b8/3b/b83b2e834cedaad246e323cbfbe19cf8486ddccd.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1145/513819.513821"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> acm.org </button> </a>

Approximating minimum size weakly-connected dominating sets for clustering mobile ad hoc networks

Yuanzhu Peter Chen, Arthur L. Liestman
<span title="">2002</span> <i title="ACM Press"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/c4l4x7lntbdcrigp6kd7u2itqq" style="color: black;">Proceedings of the 3rd ACM international symposium on Mobile ad hoc networking &amp; computing - MobiHoc &#39;02</a> </i> &nbsp;
We present a series of approximation algorithms for finding a small weakly-connected dominating set (WCDS) in a given graph to be used in clustering mobile ad hoc networks.  ...  Comparisons between our work and some previous work (CDS-based) are also given in terms of the size of resultant dominating sets and graph connectivity degradation.  ...  Figure 7 depicts the situation after 2 iterations on the given graph. In the figure, the five circled vertices are a minimum weakly-connected dominating set (OP T ).  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1145/513800.513821">doi:10.1145/513800.513821</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/mobihoc/ChenL02.html">dblp:conf/mobihoc/ChenL02</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/xxymdjx5qnd2bcfh4gsylohcla">fatcat:xxymdjx5qnd2bcfh4gsylohcla</a> </span>
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Dominating sets whose closed stars form spanning trees

Jerrold W. Grossman
<span title="">1997</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
Among the results in this paper are a characterization of those values of n and m for which there exists a connected graph with n vertices and m edges that has no spanning star tree, and a proof that finding  ...  For a subset W of vertices of an undirected graph G, let S(W) be the subgraph consisting of W, all edges incident to at least one vertex in W, and all vertices adjacent to at least one vertex in W.  ...  Acknowledgements The author thanks Suzanne Zeitman for helpful conversations on several aspects of this paper.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0012-365x(95)00334-s">doi:10.1016/0012-365x(95)00334-s</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/i22kctg4jjdrtb5fjoebflizmm">fatcat:i22kctg4jjdrtb5fjoebflizmm</a> </span>
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Maintaining weakly-connected dominating sets for clustering ad hoc networks

Yuanzhu Peter Chen, Arthur L. Liestman
<span title="">2005</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/l3w5c2fk6bh2fiux4zxdhtupaa" style="color: black;">Ad hoc networks</a> </i> &nbsp;
The size of the weakly-connected dominating set obtained also varies depending on the choice of zone size, with larger zones generally resulting in smaller weakly-connected dominating sets.  ...  Liestman, Approximating minimum size weakly-connected dominating sets for clustering mobile ad hoc networks, in: The Third ACM International Symposium on Mobile Ad 305-322].  ...  A connected dominating set S of a given graph G is a dominating set whose induced subgraph, denoted hSi, is connected. Of primary interest here is the weakly-connected dominating set.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.adhoc.2004.08.009">doi:10.1016/j.adhoc.2004.08.009</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/moixcdcghrhqpiwsi55ceyq7tm">fatcat:moixcdcghrhqpiwsi55ceyq7tm</a> </span>
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On weakly connected domination in graphs

Jean E. Dunbar, Jerrold W. Grossman, Johannes H. Hattingh, Stephen T. Hedetniemi, Alice A. McRae
<span title="">1997</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
A weakly connected dominating set for a connected graph is a dominating set D of vertices of the graph such that the edges not incident to any vertex in D do not separate the graph.  ...  This paper considers the weakly connected domination number, 7w, and related domination parameters. It is shown that the problem of computing 7w is NP-hard in general but linear for trees.  ...  It is easy to show by induction on the order of the graph that every connected graph has a weakly connected independent dominating set.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0012-365x(96)00233-6">doi:10.1016/s0012-365x(96)00233-6</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/wlytrdqagjeapdo3z6lnva7rxy">fatcat:wlytrdqagjeapdo3z6lnva7rxy</a> </span>
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On the connected and weakly convex domination numbers [article]

Magda Dettlaff, Magdalena Lemańska, Dorota Osula, María José Souto-Salorio
<span title="2019-02-20">2019</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We show that in general the difference between these numbers can be arbitrarily large and we focus on the graphs for which a weakly convex domination number equals a connected domination number.  ...  We also study the influence of the edge removing on the weakly convex domination number, in particular we prove that the weakly convex domination number is an interpolating function.  ...  Section 4 is devoted to study the influence of the edge removing on the weakly convex domination number, in particular we show that a weakly convex domination number is an interpolating function.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1902.07505v1">arXiv:1902.07505v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/musaxva3xnbndgymsv5bkml6jm">fatcat:musaxva3xnbndgymsv5bkml6jm</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20191023021732/https://arxiv.org/pdf/1902.07505v1.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/60/28/60285aa7a8c377fe60527b497697fe581a5d1135.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1902.07505v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Restrained weakly connected independent domination in the join of graphs

Rene E. Leonida
<span title="">2015</span> <i title="Hikari, Ltd."> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/tkwosg7ldfdb7axzd4gooi4inq" style="color: black;">Applied Mathematical Sciences</a> </i> &nbsp;
A connected graph is constructed with a given weakly connected independent domination number, restrained weakly connected independent domination number, and maximum weakly connected independent domination  ...  In particular, we characterized the restrained weakly connected independent dominating sets in the join of graphs and obtain the restrained weakly connected independent domination numbers.  ...  A subset S of V (G) is called weakly connected if the subgraph S w = (N G [S], E W ) weakly induced by S, is connected, where E W is the set of all edges with at least one vertex in S.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.12988/ams.2015.4121045">doi:10.12988/ams.2015.4121045</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/i6rr2nr7hjbbthtnfgid2els7e">fatcat:i6rr2nr7hjbbthtnfgid2els7e</a> </span>
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