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On Bondage Numbers of Graphs: A Survey with Some Comments

Jun-Ming Xu
<span title="">2013</span> <i title="Hindawi Limited"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/f72dp5eccrhtldziqtnbtph53m" style="color: black;">International Journal of Combinatorics</a> </i> &nbsp;
The bondage number of a nonempty graph is the smallest number of edges whose removal from results in a graph with domination number greater than the domination number of .  ...  The domination number of a graph is the smallest number of vertices which dominate all remaining vertices by edges of .  ...  Acknowledgments This work was supported by NNSF of China (Grants nos. 11071233, 61272008).  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1155/2013/595210">doi:10.1155/2013/595210</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/6zg7jz7i5jenzc2e2izfdy2txq">fatcat:6zg7jz7i5jenzc2e2izfdy2txq</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170810142747/http://staff.ustc.edu.cn/~xujm/201308.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/5d/8a/5d8a3d23590b41c34ca5427d404b42a89d797c49.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1155/2013/595210"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> hindawi.com </button> </a>

Some More Results on Total Equitable Bondage Number of A Graph

S. K. Vaidya, A. D. Parmar
<span title="2019-09-01">2019</span> <i title="Bangladesh Journals Online (JOL)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/7lpoa5kl3je4bpzjmtj6mqhdza" style="color: black;">Journal of Scientific Research</a> </i> &nbsp;
The bondage number b(G) of a nonempty graph G is the minimum cardinality among all sets of edges E0 ⊆ E(G) for which γ(G-E0) > γ (G).  ...  If γte(G) ≠ |V(G)| and <G-E0> contains no isolated vertices then the total equitable bondage number bte(G) of a graph G is the minimum cardinality among all sets of edges E0 ⊆ E(G) for which γte(G-E0)  ...  Acknowledgment The authors are highly thankful to the anonymous referee for kind comments and constructive suggestions on the first draft of this paper.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.3329/jsr.v11i3.40573">doi:10.3329/jsr.v11i3.40573</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/frqewmlbgvc6pa6aqpvmvo3qlq">fatcat:frqewmlbgvc6pa6aqpvmvo3qlq</a> </span>
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The Kp - Bondage And Kp - Non Bondage Number Of Fuzzy Graphs And Graceful Graph

R. Jahir Hussain, R. M. Karthik Keyan
<span title="">2017</span> <i title="IOSR Journals"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/pjqoxn5csbfqtdtc6supcdfpfm" style="color: black;">IOSR Journal of Electrical and Electronics Engineering</a> </i> &nbsp;
In this paper, we define the bondage b k (G) ,Co-bondage b kc (G), and Non-bondageb kn (G) number for any fuzzy graph and exact values for some standard graphs are found and some bounds are obtained.  ...  Keywords: Minimum dominating set K (G), maximum non-bondage number b kn (G),minimum bondage number b k (G).  ...  The independent domination number i(G) of a graph G is the minimum cardinality of an independent dominating set.since independent set is dominate set so (G)≤ i(G).Definition 8.2.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.9790/1676-1203051020">doi:10.9790/1676-1203051020</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/lfl3fa2i4fbyhibg3ktr7gugji">fatcat:lfl3fa2i4fbyhibg3ktr7gugji</a> </span>
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BONDAGE AND NON-BONDAGE NUMBER OF A FUZZY GRAPH

A.N. Gani, K.P. Devi, M. Akram
<span title="2015-08-06">2015</span> <i title="Academic Publications"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/mfso7il7nvhjhmry5q2brdie5m" style="color: black;">International Journal of Pure and Applied Mathematics</a> </i> &nbsp;
In this paper, bondage and non-bondage set of a fuzzy graph are discussed. The bondage number b(G) and non-bondage number b n (G) of a fuzzy graph G are defined.  ...  The exact values of b(G) and b n (G) are determined for several classes of fuzzy graphs.  ...  If G is a complete fuzzy graph with n vertices or nodes then b n (G) = (n − 1)(n − 2)/2. Proof. Let G be a complete fuzzy graph with n vertices. In G, all arcs are strong arcs.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.12732/ijpam.v103i2.7">doi:10.12732/ijpam.v103i2.7</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ts2c6gwc7ngknh5tdhwth2fnvi">fatcat:ts2c6gwc7ngknh5tdhwth2fnvi</a> </span>
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On Bondage Numbers of Graphs -- a survey with some comments [article]

Jun-Ming Xu
<span title="2012-04-18">2012</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
The bondage number of a nonempty graph G is the cardinality of a smallest edge set whose removal from G results in a graph with domination number greater than the domination number of G.  ...  This lecture gives a survey on the bondage number, including the known results, problems and conjectures. We also summarize other types of bondage numbers.  ...  bounds of the bondage number of a graph.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1204.4010v1">arXiv:1204.4010v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/px2cuvbi6fhl5ci6hpoz5fyh3y">fatcat:px2cuvbi6fhl5ci6hpoz5fyh3y</a> </span>
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The bondage number of a graph

John Frederick Fink, Michael S. Jacobson, Lael F. Kinch, John Roberts
<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;
We define the bondage number b(G) of a graph G to be the cardinality of a smallest set E of edges for which a(G -E) > a(G).  ...  A set D of vertices in a graph G is a dominating set if each vertex of G that is not in D is adjacent to at least one vertex of D.  ...  General bounds In this section we shall establish bounds on the bondage number of a graph that are independent of the graph's structure.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0012-365x(90)90348-l">doi:10.1016/0012-365x(90)90348-l</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/sbbg47vvoreenkweoylyw4ec7m">fatcat:sbbg47vvoreenkweoylyw4ec7m</a> </span>
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Upper bounds on the bondage number of a graph

Vladimir Dimitrov Samodivkin
<span title="2018-04-03">2018</span> <i title="The Institute for Research and Community Services (LPPM) ITB"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/lpemckh6jfhehiaagnsspj5vmu" style="color: black;">Electronic Journal of Graph Theory and Applications</a> </i> &nbsp;
The bondage number b(G) of a graph G is the smallest number of edges whose removal from G results in a graph with larger domination number.  ...  As a corollary we give a stronger bound than the known constant upper bounds for the bondage number of graphs having domination number at least four. Several unanswered questions are posed.  ...  The independence number β 0 (G) of a graph G is the size of the largest independent set in G.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.5614/ejgta.2018.6.1.1">doi:10.5614/ejgta.2018.6.1.1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/6ohhgnnxb5gs7hblypg4j2xomu">fatcat:6ohhgnnxb5gs7hblypg4j2xomu</a> </span>
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The bondage number of chordal graphs [article]

Valentin Bouquet
<span title="2022-03-17">2022</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
The bondage number b(G) of a graph G is the smallest cardinality of a set edges A⊆ E(G) such that γ(G-A)=γ(G)+1. A chordal graph is a graph with no induced cycle of length four or more.  ...  In this paper, we prove that the bondage number of a chordal graph G is at most the order of its maximum clique, that is, b(G)≤ω(G). We show that this bound is best possible.  ...  The bondage number b(G) of a graph G is the minimum number of edges whose removal from G increases the domination number, that is, with E ′ ⊆ E(G) such that γ(G − E ′ ) = γ(G) + 1.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2203.09256v1">arXiv:2203.09256v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/umsvmzcb6vcdzaz4qzklr72c7q">fatcat:umsvmzcb6vcdzaz4qzklr72c7q</a> </span>
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Total Equitable Bondage Number of a Graph

S. K. Vaidya, A. D. Parmar
<span title="2018-09-01">2018</span> <i title="Bangladesh Journals Online (JOL)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/7lpoa5kl3je4bpzjmtj6mqhdza" style="color: black;">Journal of Scientific Research</a> </i> &nbsp;
The bondage number b(G) of a nonempty graph G is the minimum cardinality among all sets of edges E<sub>0</sub> ⊆ E(G) for which γ(G – E<sub>0</sub>) &gt; γ(G).  ...  We introduced the concept of total equitable bondage number and proved several results.</p>  ...  The total equitable bondage number is a bondage number with the additional property that removal of an edge subset from the given graph results in a graph with larger total equitable domination number.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.3329/jsr.v10i3.33940">doi:10.3329/jsr.v10i3.33940</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/lqnctxvqubchvbjflaaqcyo6mq">fatcat:lqnctxvqubchvbjflaaqcyo6mq</a> </span>
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Page 638 of Mathematical Reviews Vol. , Issue 94b [page]

<span title="">1994</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;
The total bondage number of a graph. (English summary) Advances in graph theory, 227-235, Vishwa, Gulbarga, 1991.  ...  Summary: “We consider the problem of approximating the size of a minimum nonextendible independent set of a graph, also known as the minimum dominating independence number.  ... 
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Bondage Number of Lexicographic Product of Two Graphs

<span title="2019-07-10">2019</span> <i title="Blue Eyes Intelligence Engineering and Sciences Engineering and Sciences Publication - BEIESP"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/cj3bm7tgcffurfop7xzswxuks4" style="color: black;">VOLUME-8 ISSUE-10, AUGUST 2019, REGULAR ISSUE</a> </i> &nbsp;
The bondage number b(G) of a nonempty graph G is the cardinality of a smallest set of edges whose removal from G results in a graph with a domination number greater than the domination number of G.  ...  In this paper, we study the bondage number of the Lexicographic product of two paths, Lexicographic product of path and a graph with given maximum degree.  ...  The bondage number b(G) of a nonempty graph G is the cardinality of a smallest set of edges whose removal from G results in a graph with domination number greater than γ(G). In 1990, Fink et al.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.35940/ijitee.i8521.078919">doi:10.35940/ijitee.i8521.078919</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/j7n4erpwgbagtat5b5sbefc36a">fatcat:j7n4erpwgbagtat5b5sbefc36a</a> </span>
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ON THE ROMAN BONDAGE NUMBER OF A GRAPH

A. BAHREMANDPOUR, FU-TAO HU, S. M. SHEIKHOLESLAMI, JUN-MING XU
<span title="">2013</span> <i title="World Scientific Pub Co Pte Lt"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/7hf4dgg765bxvcgmsvotzucumu" style="color: black;">Discrete Mathematics, Algorithms and Applications (DMAA)</a> </i> &nbsp;
The Roman bondage number bR(G) of a graph G with maximum degree at least two is the minimum cardinality of all sets E ⊆ E(G) for which γR(G − E ) > γR (G).  ...  A Roman dominating function on a graph The minimum weight of a Roman dominating function on a graph G is called the Roman domination number, denoted by γR(G).  ...  Bounds on the Roman bondage number In this section we establish bounds on the Roman bondage number of a graph that are independent of the graph structure. Theorem 2.1.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1142/s1793830913500018">doi:10.1142/s1793830913500018</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/dsey5qr5gzdrxpfkmwcpiyfrze">fatcat:dsey5qr5gzdrxpfkmwcpiyfrze</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190218081002/http://pdfs.semanticscholar.org/0a1b/a178007832ad9955d735684477e311f13b39.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/0a/1b/0a1ba178007832ad9955d735684477e311f13b39.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1142/s1793830913500018"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> worldscientific.com </button> </a>

The bondage number of graphs on topological surfaces: degree-S vertices and the average degree [article]

Vladimir Samodivkin
<span title="2013-05-24">2013</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
The bondage number b(G) of a graph G is the smallest number of edges whose removal from G results in a graph with larger domination number.  ...  We also present upper bounds for the bondage number of graphs in terms of the girth, domination number and Euler characteristic.  ...  on the bondage number of a graph.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1305.5692v1">arXiv:1305.5692v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/amdhkedoebc57bh636ylwriwji">fatcat:amdhkedoebc57bh636ylwriwji</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200910094805/https://arxiv.org/pdf/1305.5692v1.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/e3/91/e391dd2b20d62fc0873474ce6f6b421ce66eaf77.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1305.5692v1" 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>

Bondage Number of 1-Planar Graph

Qiaoling Ma, Sumei Zhang, Jihui Wang
<span title="">2010</span> <i title="Scientific Research Publishing, Inc,"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/t35nyaf3lzb6lei7onyd4hnf4e" style="color: black;">Applied Mathematics</a> </i> &nbsp;
The bondage number of a nonempty graph G is the cardinality of a smallest set of edges whose removal from G results in a graph a domination number greater than the domination number of G.  ...  In this paper, we prove that for a 1-planar graph G.  ...  The bondage number ) (G b of a nonempty graph G is the cardinality of a smallest set of edges whose removal from G results in a graph with domination number greater than ) (G  .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4236/am.2010.12013">doi:10.4236/am.2010.12013</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/afl6awbulfdr3hdbfxqkmoldzm">fatcat:afl6awbulfdr3hdbfxqkmoldzm</a> </span>
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On the complexity of the outer-connected bondage and the outer-connected reinforcement problems [article]

M. Hashemipour, M. R. Hooshmandasl, A. Shakiba
<span title="2018-02-02">2018</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We define the outer-connected bondage number of a graph G as the minimum number of edges whose removal from G results in a graph with an outer-connected domination number larger than the one for G.  ...  Also, the exact values of the bondage number are determined for several classes of graphs.  ...  The bondage number of G, denoted by b(G), is the minimum number of edges whose removal from G results in a graph with a domination number larger than the one for G.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1802.00649v1">arXiv:1802.00649v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ac426ebh2vd23agfz2sr46lgte">fatcat:ac426ebh2vd23agfz2sr46lgte</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200907045948/https://arxiv.org/pdf/1802.00649v1.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/d2/78/d278bebe226ebb4a37a2cf9113a9f2a19374d8a1.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1802.00649v1" 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>
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