Characterization of Some Claw-Free Graphs in Co-Secure Domination Number
For a vertex subset <i>S</i> of a graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>,</mo></mrow></semantics></math></inline-formu...
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2025-07-01
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| author | Yuexin Zhang Jiayuan Zhang Siwen Jing Xiaodong Chen Liming Xiong |
| author_facet | Yuexin Zhang Jiayuan Zhang Siwen Jing Xiaodong Chen Liming Xiong |
| author_sort | Yuexin Zhang |
| collection | DOAJ |
| description | For a vertex subset <i>S</i> of a graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>,</mo></mrow></semantics></math></inline-formula> if each vertex of <i>G</i> is either in <i>S</i> or adjacent to some vertex in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mo>,</mo></mrow></semantics></math></inline-formula> then <i>S</i> is a dominating set of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>.</mo></mrow></semantics></math></inline-formula> Let <i>S</i> be a dominating set of a graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>.</mo></mrow></semantics></math></inline-formula> If each vertex <i>v</i> not in <i>S</i> has a neighbor <i>u</i> in <i>S</i> such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>S</mi><mo>\</mo><mo>{</mo><mi>u</mi><mo>}</mo><mo>)</mo><mo>∪</mo><mo>{</mo><mi>v</mi><mo>}</mo></mrow></semantics></math></inline-formula> is also a dominating set of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>,</mo></mrow></semantics></math></inline-formula> then <i>S</i> is a secure dominating set of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>.</mo></mrow></semantics></math></inline-formula> If each vertex <i>u</i> in <i>S</i> has a neighbor <i>v</i> not in <i>S</i> such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>S</mi><mo>\</mo><mo>{</mo><mi>u</mi><mo>}</mo><mo>)</mo><mo>∪</mo><mo>{</mo><mi>v</mi><mo>}</mo></mrow></semantics></math></inline-formula> is also a dominating set of <i>G</i>, then <i>S</i> is a co-secure dominating set of <i>G</i>. The minimum cardinality of a secure (resp. co-secure) dominating set of <i>G</i> is the secure (resp. co-secure) domination number of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>.</mo></mrow></semantics></math></inline-formula> Arumugam et al. proposed the questions to characterize a graph <i>G</i> such that the co-secure domination number of <i>G</i> equals the independence number and the secure domination number of <i>G</i>, respectively. Inspired by those questions, in this paper, we obtain two classes of claw-free graphs such that the co-secure domination number equal the independence number and the secure domination number. Our results provide some theoretical basis of claw-free graphs for networks. |
| format | Article |
| id | doaj-art-e0a3820c88864624aa88ce93eda88c98 |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-07-01 |
| publisher | MDPI AG |
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| series | Mathematics |
| spelling | doaj-art-e0a3820c88864624aa88ce93eda88c982025-08-20T03:36:34ZengMDPI AGMathematics2227-73902025-07-011315242610.3390/math13152426Characterization of Some Claw-Free Graphs in Co-Secure Domination NumberYuexin Zhang0Jiayuan Zhang1Siwen Jing2Xiaodong Chen3Liming Xiong4School of Mathematics, Liaoning Normal University, Dalian 116029, ChinaSchool of Mathematics, Liaoning Normal University, Dalian 116029, ChinaSchool of Mathematics, Liaoning Normal University, Dalian 116029, ChinaSchool of Mathematics, Liaoning Normal University, Dalian 116029, ChinaSchool of Mathematics and Statistics, Beijing Key Laboratory on MCAACI, Beijing Institute of Technology, Beijing 100081, ChinaFor a vertex subset <i>S</i> of a graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>,</mo></mrow></semantics></math></inline-formula> if each vertex of <i>G</i> is either in <i>S</i> or adjacent to some vertex in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mo>,</mo></mrow></semantics></math></inline-formula> then <i>S</i> is a dominating set of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>.</mo></mrow></semantics></math></inline-formula> Let <i>S</i> be a dominating set of a graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>.</mo></mrow></semantics></math></inline-formula> If each vertex <i>v</i> not in <i>S</i> has a neighbor <i>u</i> in <i>S</i> such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>S</mi><mo>\</mo><mo>{</mo><mi>u</mi><mo>}</mo><mo>)</mo><mo>∪</mo><mo>{</mo><mi>v</mi><mo>}</mo></mrow></semantics></math></inline-formula> is also a dominating set of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>,</mo></mrow></semantics></math></inline-formula> then <i>S</i> is a secure dominating set of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>.</mo></mrow></semantics></math></inline-formula> If each vertex <i>u</i> in <i>S</i> has a neighbor <i>v</i> not in <i>S</i> such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>S</mi><mo>\</mo><mo>{</mo><mi>u</mi><mo>}</mo><mo>)</mo><mo>∪</mo><mo>{</mo><mi>v</mi><mo>}</mo></mrow></semantics></math></inline-formula> is also a dominating set of <i>G</i>, then <i>S</i> is a co-secure dominating set of <i>G</i>. The minimum cardinality of a secure (resp. co-secure) dominating set of <i>G</i> is the secure (resp. co-secure) domination number of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>.</mo></mrow></semantics></math></inline-formula> Arumugam et al. proposed the questions to characterize a graph <i>G</i> such that the co-secure domination number of <i>G</i> equals the independence number and the secure domination number of <i>G</i>, respectively. Inspired by those questions, in this paper, we obtain two classes of claw-free graphs such that the co-secure domination number equal the independence number and the secure domination number. Our results provide some theoretical basis of claw-free graphs for networks.https://www.mdpi.com/2227-7390/13/15/2426co-secure dominating setsecure dominating setco-secure domination numbersecure domination number |
| spellingShingle | Yuexin Zhang Jiayuan Zhang Siwen Jing Xiaodong Chen Liming Xiong Characterization of Some Claw-Free Graphs in Co-Secure Domination Number Mathematics co-secure dominating set secure dominating set co-secure domination number secure domination number |
| title | Characterization of Some Claw-Free Graphs in Co-Secure Domination Number |
| title_full | Characterization of Some Claw-Free Graphs in Co-Secure Domination Number |
| title_fullStr | Characterization of Some Claw-Free Graphs in Co-Secure Domination Number |
| title_full_unstemmed | Characterization of Some Claw-Free Graphs in Co-Secure Domination Number |
| title_short | Characterization of Some Claw-Free Graphs in Co-Secure Domination Number |
| title_sort | characterization of some claw free graphs in co secure domination number |
| topic | co-secure dominating set secure dominating set co-secure domination number secure domination number |
| url | https://www.mdpi.com/2227-7390/13/15/2426 |
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